| 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 |
| 108.2-a1 |
108.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{30} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{3} \) |
$2.083836014$ |
$1.691075011$ |
4.850660303 |
\( -\frac{4821573294500}{387420489} a - \frac{20664788341000}{387420489} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -11315 a - 49303\) , \( 1367816 a + 5962187\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-11315a-49303\right){x}+1367816a+5962187$ |
| 108.2-a2 |
108.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{30} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$6.251508043$ |
$0.563691670$ |
4.850660303 |
\( \frac{4821573294500}{387420489} a - \frac{20664788341000}{387420489} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 2147734 a + 9361767\) , \( 3399421205 a + 14817733510\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(2147734a+9361767\right){x}+3399421205a+14817733510$ |
| 108.2-a3 |
108.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{18} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$3.125754021$ |
$2.254766681$ |
4.850660303 |
\( -\frac{26495862736000}{19683} a + \frac{115492849250000}{19683} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -755841 a - 3294623\) , \( 496839114 a + 2165671500\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-755841a-3294623\right){x}+496839114a+2165671500$ |
| 108.2-a4 |
108.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{18} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{3} \) |
$1.041918007$ |
$6.764300045$ |
4.850660303 |
\( \frac{26495862736000}{19683} a + \frac{115492849250000}{19683} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -11340 a - 49413\) , \( 1361411 a + 5934266\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-11340a-49413\right){x}+1361411a+5934266$ |
| 108.2-b1 |
108.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{14} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2^{3} \cdot 3 \) |
$0.062607900$ |
$13.89211744$ |
4.788858738 |
\( -\frac{65536}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -2278 a - 9920\) , \( 216157 a + 942198\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2278a-9920\right){x}+216157a+942198$ |
| 108.2-c1 |
108.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{8} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.023419254$ |
$26.94661052$ |
3.474654771 |
\( -\frac{7521280}{9} a - \frac{32817152}{9} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( a - 4\) , \( -a + 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a-4\right){x}-a+2$ |
| 108.2-c2 |
108.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.210773294$ |
$8.982203508$ |
3.474654771 |
\( -\frac{117760}{729} a + \frac{514048}{729} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -909 a + 3966\) , \( 13172 a - 57424\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-909a+3966\right){x}+13172a-57424$ |
| 108.2-d1 |
108.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{14} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$2.599829805$ |
1.192883725 |
\( -\frac{65536}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -2278 a - 9920\) , \( -216158 a - 942208\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2278a-9920\right){x}-216158a-942208$ |
| 108.2-e1 |
108.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{30} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \) |
$1$ |
$0.563691670$ |
1.163877644 |
\( -\frac{4821573294500}{387420489} a - \frac{20664788341000}{387420489} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -11313 a - 49316\) , \( -1401757 a - 6110119\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-11313a-49316\right){x}-1401757a-6110119$ |
| 108.2-e2 |
108.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{8} \cdot 3^{30} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$1.691075011$ |
1.163877644 |
\( \frac{4821573294500}{387420489} a - \frac{20664788341000}{387420489} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( 2147738 a + 9361774\) , \( -3392977996 a - 14789648202\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(2147738a+9361774\right){x}-3392977996a-14789648202$ |
| 108.2-e3 |
108.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{18} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$6.764300045$ |
1.163877644 |
\( -\frac{26495862736000}{19683} a + \frac{115492849250000}{19683} \) |
\( \bigl[a + 1\) , \( -1\) , \( 0\) , \( -755837 a - 3294616\) , \( -499106630 a - 2175555362\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-755837a-3294616\right){x}-499106630a-2175555362$ |
| 108.2-e4 |
108.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{18} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2 \) |
$1$ |
$2.254766681$ |
1.163877644 |
\( \frac{26495862736000}{19683} a + \frac{115492849250000}{19683} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -11338 a - 49426\) , \( -1395427 a - 6082528\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-11338a-49426\right){x}-1395427a-6082528$ |
| 108.2-f1 |
108.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{8} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{2} \) |
$1.556780194$ |
$2.010119168$ |
2.871655205 |
\( -\frac{7521280}{9} a - \frac{32817152}{9} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( a - 4\) , \( -12\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(a-4\right){x}-12$ |
| 108.2-f2 |
108.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{19}) \) |
$2$ |
$[2, 0]$ |
108.2 |
\( 2^{2} \cdot 3^{3} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.51132$ |
$(-3a+13), (-a-4), (-a+4)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.518926731$ |
$6.030357505$ |
2.871655205 |
\( -\frac{117760}{729} a + \frac{514048}{729} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -909 a + 3966\) , \( -13173 a + 57414\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-909a+3966\right){x}-13173a+57414$ |
*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.
