By Anthony Salvanto, Jennifer De Pinto, Sarah Dutton and Fred Backus

Heading into the early primary contests, Donald Trump continues to hold a double digit lead over the rest of the Republican field nationally. Thirty-six percent of Republican primary voters support Trump, 17 points ahead of his closest rival, Texas Senator Ted Cruz (19 percent). Florida Senator Marco Rubio has now climbed to third place at 12 percent. Ben Carson -- who briefly surged to the top of the field at the end of October -- is tied with former Florida Governor Jeb Bush at 6 percent.

Trump voters are firmer in their support for their candidate than are Republican primary voters who are voting for someone else. While 52 percent of Trump voters say their mind is made up, this is true of just 24 percent of Republican primary voters who support another candidate.

Trump is the second choice of many Republican primary voters who don't support him. Eighteen percent of Republican primary voters who are not supporting Trump pick him as their top second choice, and 17 percent choose Marco Rubio. Trump voters overwhelmingly choose Ted Cruz (31 percent) as their top second choice candidate.

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Enthusiasm for the Eventual Nominee

Enthusiasm for both Donald Trump and Ted Cruz has grown considerably since December. Now 40 percent of Republican primary voters say they would enthusiastically support Trump if he became the Republican nominee, up from 29 percent in December. Similarly, 37 percent say they would enthusiastically support Ted Cruz, up from 26 percent at the end of last year. Enthusiasm for Marco Rubio (32 percent) remains virtually unchanged.

Most Republican primary voters expect Trump to be the party's nominee. Fifty-three percent think he'll win the nomination, far ahead of any other candidate.

Candidate Qualities

More than six in 10 Republican primary voters have at least some confidence that each of the three leading candidates in the Republican race - Donald Trump, Ted Cruz, and Marco Rubio - will be an effective commander-in- chief of the nation's military. Trump has a slight edge over Cruz: 34 percent of Republican primary voters are very confident in Trump, while 30 percent are very confident in Cruz. Republican primary voters show less confidence in Marco Rubio: just 20 percent are very confident that he will be an effective commander- in-chief.

While majorities of Republican primary voters think both Cruz (68 percent) and Rubio (64 percent) have the right temperament to be President, they are more divided over Trump (50 percent).

Although about six in10 Republican primary voters think all three candidates share their values, slightly more think so of Cruz (63 percent) than either Trump (59 percent) or Rubio (59 percent).

The poll reflects dissatisfaction among Republican primary voters. Over seven in 10 Republican primary voters think America's culture and values are changing in a direction that they disagree with, far more than Democratic primary voters (40 percent) or Americans overall (54 percent). Trump voters (78 percent) are more likely to disagree with the change in the country's values and culture than Republican primary voters who support other candidates (71 percent).

Issues: The 2010 Affordable Care Act

Republican primary voters continue to strongly oppose the 2010 Affordable Care Act in far greater measure than Americans overall. Eighty-eight percent of Republican primary voters disapprove of the law, and about 73 percent say they disapprove strongly.

Repealing the 2010 Affordable Care Act is popular with Republican primary voters, and it has been advocated by all of the candidates running for the Republican nomination. While 53 percent of Americans think just minor changes are needed to make the current law work better, 65 percent of Republican primary voters think the law has so much wrong with it that it needs to be repealed entirely.

Issues: Illegal Immigration

While Republican primary voters overall are divided over whether to allow undocumented immigrants already living in the U.S. to stay in some fashion, 59 percent of Trump supporters think all undocumented immigrants in the U.S. should be required to leave the country.

Most Americans overall think immigrants here illegally should be allowed to stay in the country and apply for citizenship.

And while building a wall along the U.S.-Mexico border finds favor with Republican primary voters in general (75 percent), it is particularly popular among Trump voters (91 percent). Americans overall are divided on this issue.

The 2016 Presidential Campaign: Attention, Enthusiasm, and Interest

Republican primary voters are paying more attention to the 2016 race for President than their Democratic counterparts. While 57 percent of Republican primary voters say they are paying a lot of attention, this is true of just 45 percent of Democratic primary voters.

Sixty-five percent of registered voters are enthusiastic about the 2016 presidential campaign, and Republican primary voters are more enthusiastic than their Democratic counterparts. Forty-seven percent of Republican primary voters say they are very enthusiastic, compared to 32 percent of Democratic primary voters.

Overall, registered voters, and particularly Republican primary voters, say the 2016 presidential campaign is interesting to them. Seventy-nine percent of Republican primary voters, compared to 67 percent of Democratic primary voters, would describe the race as interesting rather than dull.

This poll was conducted by telephone January 7-10, 2016 among a random sample of 1,276 adults nationwide, including 1,094 registered voters. Data collection was conducted on behalf of CBS News and the New York Times by SSRS of Media, PA. Phone numbers were dialed from samples of both standard land-line and cell phones.

The poll employed a random digit dial methodology. For the landline sample, a respondent was randomly selected from all adults in the household. For the cell sample, interviews were conducted with the person who answered the phone.

Interviews were conducted in English and Spanish using live interviewers.

The data have been weighted to reflect U.S. Census figures on demographic variables.

The error due to sampling for results based on the entire sample could be plus or minus three percentage points. The error for subgroups may be higher and is available by request. The margin of error includes the effects of standard weighting procedures which enlarge sampling error slightly.

The margin of error for the sample of 442 Republican primary voters is 6 percentage points.

This poll release conforms to the Standards of Disclosure of the National Council on Public Polls.