| 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 |
| 578.1-a1 |
578.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{5} \cdot 17^{6} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.905259203$ |
$8.734654754$ |
3.453295554 |
\( \frac{202547839110777}{2312} a - \frac{4217180151532453}{19652} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -40 a - 250\) , \( 213 a + 1222\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-40a-250\right){x}+213a+1222$ |
| 578.1-a2 |
578.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{15} \cdot 17^{2} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.968419734$ |
$8.734654754$ |
3.453295554 |
\( \frac{1037736657}{4352} a + \frac{1256111881}{2176} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -20 a - 50\) , \( -99 a - 242\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-20a-50\right){x}-99a-242$ |
| 578.1-b1 |
578.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{5} \cdot 17^{6} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$0.758968847$ |
2.323858007 |
\( -\frac{202547839110777}{2312} a - \frac{4217180151532453}{19652} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -3057 a - 7488\) , \( -141756 a - 347236\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-3057a-7488\right){x}-141756a-347236$ |
| 578.1-b2 |
578.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{15} \cdot 17^{2} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$6.830719624$ |
2.323858007 |
\( -\frac{1037736657}{4352} a + \frac{1256111881}{2176} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -37 a - 88\) , \( -184 a - 452\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-37a-88\right){x}-184a-452$ |
| 578.1-c1 |
578.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$20.21098874$ |
1.375183600 |
\( \frac{3048625}{1088} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-3{x}+1$ |
| 578.1-c2 |
578.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{2} \cdot 17^{12} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.245665415$ |
1.375183600 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -113\) , \( -329\bigr] \) |
${y}^2+{x}{y}={x}^{3}-113{x}-329$ |
| 578.1-c3 |
578.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{4} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$20.21098874$ |
1.375183600 |
\( \frac{8805624625}{2312} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -43\) , \( 105\bigr] \) |
${y}^2+{x}{y}={x}^{3}-43{x}+105$ |
| 578.1-c4 |
578.1-c |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{4} \cdot 17^{6} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.245665415$ |
1.375183600 |
\( \frac{120920208625}{19652} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -103\) , \( -411\bigr] \) |
${y}^2+{x}{y}={x}^{3}-103{x}-411$ |
| 578.1-d1 |
578.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{12} \cdot 17^{2} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$0.562629009$ |
$13.90059457$ |
1.596429988 |
\( \frac{3048625}{1088} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 60 a - 150\) , \( 258 a - 634\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(60a-150\right){x}+258a-634$ |
| 578.1-d2 |
578.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{2} \cdot 17^{12} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.375774058$ |
$3.475148644$ |
1.596429988 |
\( \frac{159661140625}{48275138} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2262 a - 5540\) , \( 62881 a + 154026\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2262a-5540\right){x}+62881a+154026$ |
| 578.1-d3 |
578.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{6} \cdot 17^{4} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.125258019$ |
$3.475148644$ |
1.596429988 |
\( \frac{8805624625}{2312} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 860 a - 2110\) , \( 21650 a - 53034\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(860a-2110\right){x}+21650a-53034$ |
| 578.1-d4 |
578.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( 2^{4} \cdot 17^{6} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.687887029$ |
$13.90059457$ |
1.596429988 |
\( \frac{120920208625}{19652} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -2062 a - 5050\) , \( 79317 a + 194286\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-2062a-5050\right){x}+79317a+194286$ |
| 578.1-e1 |
578.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{5} \cdot 17^{6} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.905259203$ |
$8.734654754$ |
3.453295554 |
\( -\frac{202547839110777}{2312} a - \frac{4217180151532453}{19652} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 39 a - 250\) , \( -213 a + 1222\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(39a-250\right){x}-213a+1222$ |
| 578.1-e2 |
578.1-e |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{15} \cdot 17^{2} \) |
$2.14648$ |
$(-a+2), (17)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$0.968419734$ |
$8.734654754$ |
3.453295554 |
\( -\frac{1037736657}{4352} a + \frac{1256111881}{2176} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 19 a - 50\) , \( 99 a - 242\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(19a-50\right){x}+99a-242$ |
| 578.1-f1 |
578.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{5} \cdot 17^{6} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$0.758968847$ |
2.323858007 |
\( \frac{202547839110777}{2312} a - \frac{4217180151532453}{19652} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 3056 a - 7488\) , \( 141756 a - 347236\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(3056a-7488\right){x}+141756a-347236$ |
| 578.1-f2 |
578.1-f |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
578.1 |
\( 2 \cdot 17^{2} \) |
\( - 2^{15} \cdot 17^{2} \) |
$2.14648$ |
$(-a+2), (17)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \cdot 5 \) |
$1$ |
$6.830719624$ |
2.323858007 |
\( \frac{1037736657}{4352} a + \frac{1256111881}{2176} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 36 a - 88\) , \( 184 a - 452\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(36a-88\right){x}+184a-452$ |
*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.
