| 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 |
| 722.1-a1 |
722.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( - 2^{5} \cdot 19^{8} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$7.011097592$ |
7.155671516 |
\( -\frac{38582007144025}{376367048} a + \frac{47222031794147}{188183524} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 17 a - 54\) , \( -52 a + 191\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(17a-54\right){x}-52a+191$ |
| 722.1-a2 |
722.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{4} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$14.02219518$ |
7.155671516 |
\( \frac{9618774071}{109744} a + \frac{47511875911}{219488} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -3 a - 14\) , \( 4 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-3a-14\right){x}+4a+7$ |
| 722.1-b1 |
722.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{6} \cdot 19^{2} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$32.17041206$ |
1.459279525 |
\( -\frac{413493625}{152} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -16\) , \( 22\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-16{x}+22$ |
| 722.1-b2 |
722.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{54} \cdot 19^{2} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$0.397165580$ |
1.459279525 |
\( -\frac{69173457625}{2550136832} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -86\) , \( -2456\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-86{x}-2456$ |
| 722.1-b3 |
722.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{18} \cdot 19^{6} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$3.574490228$ |
1.459279525 |
\( \frac{94196375}{3511808} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 9\) , \( 90\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+9{x}+90$ |
| 722.1-c1 |
722.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{4} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.780731450$ |
$6.502561468$ |
2.363617922 |
\( -\frac{9618774071}{109744} a + \frac{47511875911}{219488} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -163 a - 399\) , \( -358 a - 877\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-163a-399\right){x}-358a-877$ |
| 722.1-c2 |
722.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( - 2^{5} \cdot 19^{8} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.561462901$ |
$1.625640367$ |
2.363617922 |
\( \frac{38582007144025}{376367048} a + \frac{47222031794147}{188183524} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 317 a - 777\) , \( 16750 a - 41029\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(317a-777\right){x}+16750a-41029$ |
| 722.1-d1 |
722.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{2} \cdot 19^{10} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$5.550558930$ |
2.266006194 |
\( -\frac{37966934881}{4952198} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( 1399 a - 3431\) , \( -47575 a + 116533\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(1399a-3431\right){x}-47575a+116533$ |
| 722.1-d2 |
722.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{2} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$1$ |
$5.550558930$ |
2.266006194 |
\( -\frac{1}{608} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -1\) , \( -235 a - 577\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}-{x}-235a-577$ |
| 722.1-e1 |
722.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{4} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$14.02219518$ |
7.155671516 |
\( -\frac{9618774071}{109744} a + \frac{47511875911}{219488} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( 2 a - 14\) , \( -4 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2a-14\right){x}-4a+7$ |
| 722.1-e2 |
722.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( - 2^{5} \cdot 19^{8} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$7.011097592$ |
7.155671516 |
\( \frac{38582007144025}{376367048} a + \frac{47222031794147}{188183524} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 1\) , \( -18 a - 54\) , \( 52 a + 191\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-18a-54\right){x}+52a+191$ |
| 722.1-f1 |
722.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{2} \cdot 19^{10} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$3.759911303$ |
$0.671163407$ |
2.060441272 |
\( -\frac{37966934881}{4952198} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -70\) , \( -279\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-70{x}-279$ |
| 722.1-f2 |
722.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{2} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$0.751982260$ |
$16.77908518$ |
2.060441272 |
\( -\frac{1}{608} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+1$ |
| 722.1-g1 |
722.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{6} \cdot 19^{2} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1.698405069$ |
$1.446313524$ |
6.016990833 |
\( -\frac{413493625}{152} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -311 a - 759\) , \( -4716 a - 11551\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-311a-759\right){x}-4716a-11551$ |
| 722.1-g2 |
722.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{54} \cdot 19^{2} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$1.698405069$ |
$1.446313524$ |
6.016990833 |
\( -\frac{69173457625}{2550136832} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -1711 a - 4189\) , \( 484528 a + 1186849\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-1711a-4189\right){x}+484528a+1186849$ |
| 722.1-g3 |
722.1-g |
$3$ |
$9$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{18} \cdot 19^{6} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{4} \) |
$0.566135023$ |
$1.446313524$ |
6.016990833 |
\( \frac{94196375}{3511808} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -189 a + 466\) , \( 17680 a - 43306\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-189a+466\right){x}+17680a-43306$ |
| 722.1-h1 |
722.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( - 2^{5} \cdot 19^{8} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.561462901$ |
$1.625640367$ |
2.363617922 |
\( -\frac{38582007144025}{376367048} a + \frac{47222031794147}{188183524} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -317 a - 777\) , \( -16750 a - 41029\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-317a-777\right){x}-16750a-41029$ |
| 722.1-h2 |
722.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
722.1 |
\( 2 \cdot 19^{2} \) |
\( 2^{10} \cdot 19^{4} \) |
$2.26923$ |
$(-a+2), (a+5), (a-5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.780731450$ |
$6.502561468$ |
2.363617922 |
\( \frac{9618774071}{109744} a + \frac{47511875911}{219488} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 163 a - 399\) , \( 358 a - 877\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(163a-399\right){x}+358a-877$ |
*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.
