| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 17.1-a1 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$6.070512355$ |
$0.598363940$ |
0.440490254 |
\( -\frac{97064067741644382786}{17} a + \frac{248634735746274843273}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 330 a - 936\) , \( 4996 a - 13110\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(330a-936\right){x}+4996a-13110$ |
| 17.1-a2 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{8} \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.758814044$ |
$2.393455763$ |
0.440490254 |
\( -\frac{35937}{83521} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( -14\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}-14$ |
| 17.1-a3 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.379407022$ |
$38.29529222$ |
0.440490254 |
\( -\frac{5821794}{17} a + \frac{14936697}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 6 a + 9\) , \( 4 a + 6\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(6a+9\right){x}+4a+6$ |
| 17.1-a4 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.758814044$ |
$38.29529222$ |
0.440490254 |
\( \frac{35937}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$ |
| 17.1-a5 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{4} \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.517628088$ |
$9.573823055$ |
0.440490254 |
\( \frac{20346417}{289} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6\) , \( -4\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-6{x}-4$ |
| 17.1-a6 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.379407022$ |
$38.29529222$ |
0.440490254 |
\( \frac{5821794}{17} a + \frac{9114903}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6 a + 15\) , \( -4 a + 10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-6a+15\right){x}-4a+10$ |
| 17.1-a7 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$3.035256177$ |
$2.393455763$ |
0.440490254 |
\( \frac{82483294977}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -91\) , \( -310\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-91{x}-310$ |
| 17.1-a8 |
17.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$0.74813$ |
$(-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$6.070512355$ |
$0.598363940$ |
0.440490254 |
\( \frac{97064067741644382786}{17} a + \frac{151570668004630460487}{17} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -330 a - 606\) , \( -4996 a - 8114\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-330a-606\right){x}-4996a-8114$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
