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 |
15876.3-a1 |
15876.3-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{10} \cdot 7^{3} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \cdot 3 \) |
$0.058798633$ |
$3.000417687$ |
2.444553594 |
\( -\frac{2271}{2} a + 945 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 3 a\) , \( -3 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+3a{x}-3a+4$ |
15876.3-a2 |
15876.3-a |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{14} \cdot 3^{10} \cdot 7^{9} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B |
$1$ |
\( 2 \cdot 3 \) |
$0.411590433$ |
$0.428631098$ |
2.444553594 |
\( \frac{15883557}{128} a + \frac{1611225}{64} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 3 a - 630\) , \( -129 a - 6044\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(3a-630\right){x}-129a-6044$ |
15876.3-b1 |
15876.3-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 7^{6} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 1 \) |
$1$ |
$1.229687320$ |
1.419920610 |
\( \frac{17268549}{2} a - \frac{246587109}{2} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 72 a - 214\) , \( 493 a - 1159\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(72a-214\right){x}+493a-1159$ |
15876.3-b2 |
15876.3-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 7^{6} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 1 \) |
$1$ |
$1.229687320$ |
1.419920610 |
\( -\frac{17268549}{2} a - 114659280 \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -194 a + 11\) , \( -1163 a + 589\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-194a+11\right){x}-1163a+589$ |
15876.3-b3 |
15876.3-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 7^{6} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 1 \) |
$1$ |
$1.229687320$ |
1.419920610 |
\( -\frac{132651}{2} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 76 a - 29\) , \( -153 a - 106\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(76a-29\right){x}-153a-106$ |
15876.3-b4 |
15876.3-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{18} \cdot 3^{4} \cdot 7^{6} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B[2] |
$1$ |
\( 1 \) |
$1$ |
$1.229687320$ |
1.419920610 |
\( -\frac{1167051}{512} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 36 a - 14\) , \( 65 a + 55\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(36a-14\right){x}+65a+55$ |
15876.3-b5 |
15876.3-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{6} \cdot 3^{12} \cdot 7^{6} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 1 \) |
$1$ |
$1.229687320$ |
1.419920610 |
\( \frac{9261}{8} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -32 a + 12\) , \( 32 a + 18\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-32a+12\right){x}+32a+18$ |
15876.3-c1 |
15876.3-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{6} \cdot 3^{12} \cdot 7^{9} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2 \) |
$1$ |
$0.749014438$ |
1.729774751 |
\( -\frac{1192725}{1372} a + \frac{287307}{2744} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -76 a + 32\) , \( -378 a + 254\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-76a+32\right){x}-378a+254$ |
15876.3-c2 |
15876.3-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 7^{15} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.749014438$ |
1.729774751 |
\( \frac{50481832659}{80707214} a + \frac{5130085995}{40353607} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 72 a - 27\) , \( -299 a + 106\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(72a-27\right){x}-299a+106$ |
15876.3-c3 |
15876.3-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{18} \cdot 3^{4} \cdot 7^{7} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.749014438$ |
1.729774751 |
\( -\frac{457190997}{1792} a + \frac{1524555639}{3584} \) |
\( \bigl[a\) , \( a\) , \( 0\) , \( 167 a - 277\) , \( 1305 a - 1549\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(167a-277\right){x}+1305a-1549$ |
15876.3-c4 |
15876.3-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{12} \cdot 7^{7} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.749014438$ |
1.729774751 |
\( -\frac{43477641}{14} a + \frac{34139475}{14} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -179 a + 374\) , \( 1805 a + 621\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-179a+374\right){x}+1805a+621$ |
15876.3-c5 |
15876.3-c |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 7^{7} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2 \) |
$1$ |
$0.749014438$ |
1.729774751 |
\( -\frac{13989364197}{7} a + \frac{30175519239}{14} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 0\) , \( 535 a - 758\) , \( 6903 a - 6086\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(535a-758\right){x}+6903a-6086$ |
15876.3-d1 |
15876.3-d |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 7^{9} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.2.1 |
$1$ |
\( 2 \) |
$0.328226405$ |
$1.964234452$ |
2.977804628 |
\( -\frac{2271}{2} a + 945 \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( -7 a + 11\) , \( -3 a + 25\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-7a+11\right){x}-3a+25$ |
15876.3-d2 |
15876.3-d |
$2$ |
$7$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
15876.3 |
\( 2^{2} \cdot 3^{4} \cdot 7^{2} \) |
\( 2^{14} \cdot 3^{4} \cdot 7^{3} \) |
$1.73734$ |
$(-2a+1), (3a-2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$7$ |
7B.2.3 |
$1$ |
\( 2 \cdot 7 \) |
$0.046889486$ |
$1.964234452$ |
2.977804628 |
\( \frac{15883557}{128} a + \frac{1611225}{64} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 20 a + 14\) , \( -46 a + 65\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(20a+14\right){x}-46a+65$ |
*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.