| 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 |
| 1764.3-a1 |
1764.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{15} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.798377478$ |
$1.338961176$ |
2.671389133 |
\( -\frac{1566386529851}{275562} a - \frac{680059534037}{91854} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -26 a + 25\) , \( -29 a + 9\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-26a+25\right){x}-29a+9$ |
| 1764.3-a2 |
1764.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 3^{9} \cdot 7^{6} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.599459159$ |
$4.016883529$ |
2.671389133 |
\( \frac{155473175}{37044} a - \frac{238701437}{24696} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -4 a - 7\) , \( -506 a - 658\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-4a-7\right){x}-506a-658$ |
| 1764.3-a3 |
1764.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{7} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1.798377478$ |
$12.05065058$ |
2.671389133 |
\( \frac{23257750547483}{21} a - \frac{35704920899449}{14} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 21 a - 41\) , \( 36 a + 235\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(21a-41\right){x}+36a+235$ |
| 1764.3-b1 |
1764.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$8.981464447$ |
2.491010045 |
\( -\frac{226981}{14} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -7 a - 9\) , \( 12 a + 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-7a-9\right){x}+12a+15$ |
| 1764.3-b2 |
1764.3-b |
$2$ |
$5$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{10} \cdot 3^{6} \cdot 7^{10} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$1$ |
$1.796292889$ |
2.491010045 |
\( \frac{5735339}{537824} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( 18 a + 26\) , \( -778 a - 1001\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(18a+26\right){x}-778a-1001$ |
| 1764.3-c1 |
1764.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{14} \cdot 3^{7} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.869844602$ |
$4.387960917$ |
4.234408362 |
\( \frac{366889}{2688} a + \frac{107635}{448} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -2 a + 16\) , \( 47 a - 98\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a+16\right){x}+47a-98$ |
| 1764.3-d1 |
1764.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{7} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.119245093$ |
$21.57078626$ |
2.853611247 |
\( -\frac{25609}{21} a - \frac{24473}{14} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -2\) , \( -2 a + 4\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}-2{x}-2a+4$ |
| 1764.3-e1 |
1764.3-e |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{10} \cdot 3^{7} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \cdot 5 \) |
$0.211070099$ |
$4.065560328$ |
4.759983467 |
\( -\frac{25435927469363}{672} a - \frac{5522884821905}{112} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 126 a - 317\) , \( 1373 a - 3149\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(126a-317\right){x}+1373a-3149$ |
| 1764.3-f1 |
1764.3-f |
$2$ |
$5$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{20} \cdot 3^{9} \cdot 7^{10} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.2 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.387438780$ |
2.149123677 |
\( -\frac{1602887926644071}{464679936} a - \frac{354692639101429}{77446656} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -601 a - 2729\) , \( -37344 a - 16123\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-601a-2729\right){x}-37344a-16123$ |
| 1764.3-f2 |
1764.3-f |
$2$ |
$5$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{21} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$5$ |
5B.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.937193904$ |
2.149123677 |
\( \frac{153015935281}{401769396} a + \frac{108791805055}{133923132} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 4 a + 26\) , \( 16 a - 44\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(4a+26\right){x}+16a-44$ |
| 1764.3-g1 |
1764.3-g |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{7} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.047073196$ |
$14.36960177$ |
3.001696163 |
\( \frac{15853}{84} a - \frac{977}{14} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -1\) , \( 1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-{x}+1$ |
| 1764.3-h1 |
1764.3-h |
$6$ |
$18$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{36} \cdot 3^{6} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.010844637$ |
2.523220734 |
\( -\frac{548347731625}{1835008} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 170 a - 682\) , \( -3495 a + 6116\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(170a-682\right){x}-3495a+6116$ |
| 1764.3-h2 |
1764.3-h |
$6$ |
$18$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{4} \cdot 3^{6} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$9.097601735$ |
2.523220734 |
\( -\frac{15625}{28} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -2\) , \( a - 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}-2{x}+a-2$ |
| 1764.3-h3 |
1764.3-h |
$6$ |
$18$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{12} \cdot 3^{6} \cdot 7^{6} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.032533911$ |
2.523220734 |
\( \frac{9938375}{21952} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -5 a + 18\) , \( -23 a + 40\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-5a+18\right){x}-23a+40$ |
| 1764.3-h4 |
1764.3-h |
$6$ |
$18$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 3^{6} \cdot 7^{12} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.032533911$ |
2.523220734 |
\( \frac{4956477625}{941192} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 35 a - 142\) , \( -279 a + 488\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(35a-142\right){x}-279a+488$ |
| 1764.3-h5 |
1764.3-h |
$6$ |
$18$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{4} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$9.097601735$ |
2.523220734 |
\( \frac{128787625}{98} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 10 a - 42\) , \( 49 a - 86\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(10a-42\right){x}+49a-86$ |
| 1764.3-h6 |
1764.3-h |
$6$ |
$18$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{18} \cdot 3^{6} \cdot 7^{4} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.010844637$ |
2.523220734 |
\( \frac{2251439055699625}{25088} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 2730 a - 10922\) , \( -220583 a + 386020\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(2730a-10922\right){x}-220583a+386020$ |
| 1764.3-i1 |
1764.3-i |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{9} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$1.034584562$ |
1.147768520 |
\( \frac{1349045161}{56} a - \frac{6213102761}{112} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 46 a - 105\) , \( 221 a - 521\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(46a-105\right){x}+221a-521$ |
| 1764.3-j1 |
1764.3-j |
$1$ |
$1$ |
\(\Q(\sqrt{13}) \) |
$2$ |
$[2, 0]$ |
1764.3 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
\( 2^{8} \cdot 3^{3} \cdot 7^{2} \) |
$2.08802$ |
$(-a+1), (2), (7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.090147760$ |
$10.42672864$ |
4.171106901 |
\( \frac{1349045161}{56} a - \frac{6213102761}{112} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 8 a - 20\) , \( -16 a + 32\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(8a-20\right){x}-16a+32$ |
*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.
