| 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-a3 |
17.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$0.51321$ |
$(-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.940359004$ |
0.436670169 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -34 a - 53\) , \( -113 a - 163\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-34a-53\right){x}-113a-163$ |
| 272.1-c3 |
272.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
272.1 |
\( 2^{4} \cdot 17 \) |
\( 2^{12} \cdot 17^{3} \) |
$1.02642$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$13.44455174$ |
1.188341713 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -134 a - 208\) , \( 900 a + 1298\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-134a-208\right){x}+900a+1298$ |
| 289.3-a3 |
289.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
289.3 |
\( 17^{2} \) |
\( 17^{9} \) |
$1.04210$ |
$(-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$6.521565522$ |
1.152860801 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -64 a - 379\) , \( 693 a + 2778\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-64a-379\right){x}+693a+2778$ |
| 833.3-a3 |
833.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
833.3 |
\( 7^{2} \cdot 17 \) |
\( 7^{6} \cdot 17^{3} \) |
$1.35783$ |
$(-2a+1), (-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.344157561$ |
$7.188820246$ |
1.749443587 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( -94 a - 200\) , \( -757 a - 894\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-94a-200\right){x}-757a-894$ |
| 833.5-a3 |
833.5-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
833.5 |
\( 7^{2} \cdot 17 \) |
\( 7^{6} \cdot 17^{3} \) |
$1.35783$ |
$(2a+1), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2.639849497$ |
1.399991610 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( -a\) , \( 1\) , \( 173 a - 288\) , \( 1841 a - 2495\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(173a-288\right){x}+1841a-2495$ |
| 1377.1-a3 |
1377.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1377.1 |
\( 3^{4} \cdot 17 \) |
\( 3^{12} \cdot 17^{3} \) |
$1.53963$ |
$(-3a-1), (3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.847053893$ |
$8.963034498$ |
1.951049946 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( 1\) , \( a + 1\) , \( -302 a - 469\) , \( 3037 a + 4380\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-302a-469\right){x}+3037a+4380$ |
| 4352.1-b3 |
4352.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4352.1 |
\( 2^{8} \cdot 17 \) |
\( 2^{12} \cdot 17^{3} \) |
$2.05284$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$9.509915295$ |
2.521697097 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 14 a - 88\) , \( -190 a + 86\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(14a-88\right){x}-190a+86$ |
| 4352.1-v3 |
4352.1-v |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4352.1 |
\( 2^{8} \cdot 17 \) |
\( 2^{12} \cdot 17^{3} \) |
$2.05284$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$3.492192634$ |
0.926007409 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 14 a - 88\) , \( 190 a - 86\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(14a-88\right){x}+190a-86$ |
| 4624.3-j3 |
4624.3-j |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4624.3 |
\( 2^{4} \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{9} \) |
$2.08419$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2 \) |
$1$ |
$0.599106529$ |
0.953172651 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -256 a - 1514\) , \( -4034 a - 21720\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-256a-1514\right){x}-4034a-21720$ |
| 4913.2-a3 |
4913.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4913.2 |
\( 17^{3} \) |
\( 17^{9} \) |
$2.11602$ |
$(-3a-1), (3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.328133472$ |
$6.521565522$ |
2.269753309 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 203 a - 466\) , \( -2650 a + 4452\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(203a-466\right){x}-2650a+4452$ |
*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.
