| 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 |
| 4.1-a11 |
4.1-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{3} \) |
$0.52105$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$22.96135788$ |
0.309385960 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 11 a - 32\) , \( 27 a - 72\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(11a-32\right){x}+27a-72$ |
| 32.3-a11 |
32.3-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.3 |
\( 2^{5} \) |
\( 2^{15} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2 \) |
$1$ |
$2.914337325$ |
1.413661249 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 75 a - 209\) , \( 618 a - 1607\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(75a-209\right){x}+618a-1607$ |
| 32.4-a11 |
32.4-a |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
32.4 |
\( 2^{5} \) |
\( 2^{15} \) |
$0.87630$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$23.31469860$ |
1.413661249 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 23 a - 87\) , \( -129 a + 354\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(23a-87\right){x}-129a+354$ |
| 128.5-b11 |
128.5-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -9 a - 117\) , \( -339 a - 74\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-9a-117\right){x}-339a-74$ |
| 128.5-c11 |
128.5-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.5 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$16.48598148$ |
1.999218911 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 1656 a - 4244\) , \( -51666 a + 132346\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(1656a-4244\right){x}-51666a+132346$ |
| 128.6-b11 |
128.6-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.184918978$ |
1.014991940 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 373 a - 965\) , \( -5721 a + 14641\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+\left(373a-965\right){x}-5721a+14641$ |
| 128.6-c11 |
128.6-c |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
128.6 |
\( 2^{7} \) |
\( 2^{21} \) |
$1.23927$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$2.060747685$ |
1.999218911 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -147 a - 253\) , \( -1763 a - 2797\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-147a-253\right){x}-1763a-2797$ |
| 256.1-b11 |
256.1-b |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{27} \) |
$1.47375$ |
$(-a+2), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.959184588$ |
1.435415367 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 177 a - 524\) , \( -2112 a + 5248\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(177a-524\right){x}-2112a+5248$ |
| 324.1-e11 |
324.1-e |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
324.1 |
\( 2^{2} \cdot 3^{4} \) |
\( 2^{3} \cdot 3^{12} \) |
$1.56315$ |
$(-a+2), (-a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$3.945579451$ |
3.827774313 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 99 a - 296\) , \( -940 a + 2313\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(99a-296\right){x}-940a+2313$ |
| 676.4-i11 |
676.4-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.4 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{3} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$0.989827765$ |
$3.282920544$ |
3.152503187 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -331 a - 545\) , \( -4886 a - 7574\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-331a-545\right){x}-4886a-7574$ |
| 676.5-i11 |
676.5-i |
$12$ |
$24$ |
\(\Q(\sqrt{17}) \) |
$2$ |
$[2, 0]$ |
676.5 |
\( 2^{2} \cdot 13^{2} \) |
\( 2^{3} \cdot 13^{6} \) |
$1.87867$ |
$(-a+2), (-a-1), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1.979655531$ |
$3.282920544$ |
3.152503187 |
\( \frac{54503407609}{4} a + \frac{42555672073}{2} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -2806 a - 4383\) , \( -114963 a - 179521\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2806a-4383\right){x}-114963a-179521$ |
*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.
