| 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 |
$1$ |
$1$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$9.65055$ |
$(a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$184.4286477$ |
2.433442935 |
\( -\frac{9280666137}{4913} a^{3} - \frac{68843781023}{4913} a^{2} - \frac{24295621869}{4913} a + \frac{16652449695}{4913} \) |
\( \bigl[-a^{3} + a^{2} + 4 a\) , \( -2 a^{3} + a^{2} + 8 a + 1\) , \( -a^{3} + a^{2} + 5 a\) , \( -3 a^{3} + a^{2} + 9 a\) , \( -11 a^{3} + 7 a^{2} + 45 a - 13\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a\right){x}{y}+\left(-a^{3}+a^{2}+5a\right){y}={x}^{3}+\left(-2a^{3}+a^{2}+8a+1\right){x}^{2}+\left(-3a^{3}+a^{2}+9a\right){x}-11a^{3}+7a^{2}+45a-13$ |
| 17.1-b1 |
17.1-b |
$2$ |
$2$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{6} \) |
$9.65055$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \) |
$1$ |
$307.8024718$ |
2.030649141 |
\( -\frac{2712300915302597}{24137569} a^{3} - \frac{861583963356942}{24137569} a^{2} + \frac{13363979345271242}{24137569} a + \frac{9634906213427192}{24137569} \) |
\( \bigl[a^{2} - 1\) , \( 2 a^{3} - a^{2} - 9 a + 1\) , \( -a^{3} + a^{2} + 4 a - 1\) , \( 14 a^{3} - 2 a^{2} - 62 a - 27\) , \( 21 a^{3} + 5 a^{2} - 106 a - 67\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{3}+a^{2}+4a-1\right){y}={x}^{3}+\left(2a^{3}-a^{2}-9a+1\right){x}^{2}+\left(14a^{3}-2a^{2}-62a-27\right){x}+21a^{3}+5a^{2}-106a-67$ |
| 17.1-b2 |
17.1-b |
$2$ |
$2$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$9.65055$ |
$(a^2-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 1 \) |
$1$ |
$615.6049437$ |
2.030649141 |
\( \frac{110962134}{4913} a^{3} + \frac{199269684}{4913} a^{2} - \frac{82968178}{4913} a - \frac{107581645}{4913} \) |
\( \bigl[a + 1\) , \( a^{2} - a - 1\) , \( a^{3} - 5 a - 1\) , \( -3 a^{3} + 4 a^{2} + 12 a - 3\) , \( a^{3} + a^{2} - 6 a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{2}-a-1\right){x}^{2}+\left(-3a^{3}+4a^{2}+12a-3\right){x}+a^{3}+a^{2}-6a$ |
| 17.1-c1 |
17.1-c |
$2$ |
$3$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$9.65055$ |
$(a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$12.20518653$ |
1.449371493 |
\( -\frac{5046977834405570395}{4913} a^{3} - \frac{1470522239949015371}{4913} a^{2} + \frac{24806427686194829270}{4913} a + \frac{17321727365860720482}{4913} \) |
\( \bigl[-a^{3} + a^{2} + 4 a\) , \( a^{3} - 6 a - 1\) , \( a^{3} - 5 a - 1\) , \( 46 a^{3} - 207 a - 127\) , \( 207 a^{3} - 3 a^{2} - 915 a - 596\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-6a-1\right){x}^{2}+\left(46a^{3}-207a-127\right){x}+207a^{3}-3a^{2}-915a-596$ |
| 17.1-c2 |
17.1-c |
$2$ |
$3$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$9.65055$ |
$(a^2-2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$988.6201097$ |
1.449371493 |
\( -\frac{608242}{17} a^{3} - \frac{179684}{17} a^{2} + \frac{2976663}{17} a + \frac{2072127}{17} \) |
\( \bigl[-a^{3} + a^{2} + 4 a\) , \( a^{3} - 6 a - 1\) , \( a^{3} - 5 a - 1\) , \( a^{3} - 7 a + 3\) , \( a^{2} - 2 a - 1\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+4a\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(a^{3}-6a-1\right){x}^{2}+\left(a^{3}-7a+3\right){x}+a^{2}-2a-1$ |
| 17.1-d1 |
17.1-d |
$2$ |
$3$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( -17 \) |
$9.65055$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.051991874$ |
$713.0293279$ |
1.956571160 |
\( -\frac{608242}{17} a^{3} - \frac{179684}{17} a^{2} + \frac{2976663}{17} a + \frac{2072127}{17} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 5 a\) , \( -8 a^{3} + 5 a^{2} + 36 a - 7\) , \( 28 a^{3} - 21 a^{2} - 124 a + 36\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-8a^{3}+5a^{2}+36a-7\right){x}+28a^{3}-21a^{2}-124a+36$ |
| 17.1-d2 |
17.1-d |
$2$ |
$3$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$9.65055$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.155975624$ |
$79.22548087$ |
1.956571160 |
\( -\frac{5046977834405570395}{4913} a^{3} - \frac{1470522239949015371}{4913} a^{2} + \frac{24806427686194829270}{4913} a + \frac{17321727365860720482}{4913} \) |
\( \bigl[1\) , \( a^{3} - 5 a - 2\) , \( a^{3} - 5 a\) , \( 67 a^{3} - 45 a^{2} - 299 a + 63\) , \( -699 a^{3} + 504 a^{2} + 3111 a - 837\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(67a^{3}-45a^{2}-299a+63\right){x}-699a^{3}+504a^{2}+3111a-837$ |
| 17.1-e1 |
17.1-e |
$2$ |
$2$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{6} \) |
$9.65055$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 2 \cdot 3 \) |
$0.022379274$ |
$948.8047725$ |
1.680996848 |
\( -\frac{2712300915302597}{24137569} a^{3} - \frac{861583963356942}{24137569} a^{2} + \frac{13363979345271242}{24137569} a + \frac{9634906213427192}{24137569} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - 5 a\) , \( -23 a^{3} + 17 a^{2} + 100 a - 34\) , \( 58 a^{3} - 44 a^{2} - 257 a + 78\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-23a^{3}+17a^{2}+100a-34\right){x}+58a^{3}-44a^{2}-257a+78$ |
| 17.1-e2 |
17.1-e |
$2$ |
$2$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$9.65055$ |
$(a^2-2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3Ns |
$1$ |
\( 3 \) |
$0.044758549$ |
$948.8047725$ |
1.680996848 |
\( \frac{110962134}{4913} a^{3} + \frac{199269684}{4913} a^{2} - \frac{82968178}{4913} a - \frac{107581645}{4913} \) |
\( \bigl[-a^{3} + a^{2} + 5 a - 1\) , \( -a^{2} + 3\) , \( a^{3} - 5 a\) , \( 2 a^{3} - 3 a^{2} - 10 a + 6\) , \( -a^{3} + 4 a - 1\bigr] \) |
${y}^2+\left(-a^{3}+a^{2}+5a-1\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(2a^{3}-3a^{2}-10a+6\right){x}-a^{3}+4a-1$ |
| 17.1-f1 |
17.1-f |
$1$ |
$1$ |
4.4.5744.1 |
$4$ |
$[4, 0]$ |
17.1 |
\( 17 \) |
\( - 17^{3} \) |
$9.65055$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Ns |
$1$ |
\( 3 \) |
$0.013859577$ |
$507.6259882$ |
1.113955591 |
\( -\frac{9280666137}{4913} a^{3} - \frac{68843781023}{4913} a^{2} - \frac{24295621869}{4913} a + \frac{16652449695}{4913} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + 6 a\) , \( a^{3} - 5 a\) , \( 8 a^{3} - 12 a^{2} - 21 a + 1\) , \( -22 a^{3} + 47 a^{2} + 23 a - 23\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{3}+6a\right){x}^{2}+\left(8a^{3}-12a^{2}-21a+1\right){x}-22a^{3}+47a^{2}+23a-23$ |
*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.
