| 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 |
| 4356.6-a1 |
4356.6-a |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{7} \cdot 3^{4} \cdot 11^{7} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.171079744$ |
$1.585085287$ |
1.639918221 |
\( \frac{1375753}{6336} a - \frac{1038611}{6336} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -8 a + 4\) , \( 2 a - 34\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-8a+4\right){x}+2a-34$ |
| 4356.6-a2 |
4356.6-a |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{5} \cdot 3^{2} \cdot 11^{8} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.342159489$ |
$1.585085287$ |
1.639918221 |
\( -\frac{17031617}{2904} a + \frac{21786113}{2904} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( -12 a + 27\) , \( -17 a - 24\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-12a+27\right){x}-17a-24$ |
| 4356.6-b1 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$0.416494274$ |
$1.281706661$ |
3.228261005 |
\( \frac{4913}{1296} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -3 a + 2\) , \( 45 a + 2\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-3a+2\right){x}+45a+2$ |
| 4356.6-b2 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{17} \cdot 3^{2} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.665977099$ |
$1.281706661$ |
3.228261005 |
\( -\frac{43993943}{196608} a + \frac{140725583}{65536} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 12 a + 20\) , \( 6 a - 20\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(12a+20\right){x}+6a-20$ |
| 4356.6-b3 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{17} \cdot 3^{2} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.104123568$ |
$1.281706661$ |
3.228261005 |
\( \frac{43993943}{196608} a + \frac{189091403}{98304} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -30\) , \( 18 a + 30\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}-30{x}+18a+30$ |
| 4356.6-b4 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{4} \cdot 3^{16} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.832988549$ |
$0.640853330$ |
3.228261005 |
\( \frac{838561807}{26244} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 157 a - 138\) , \( 805 a - 58\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(157a-138\right){x}+805a-58$ |
| 4356.6-b5 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{4} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$0.208247137$ |
$1.281706661$ |
3.228261005 |
\( -\frac{56620795}{2304} a + \frac{85821697}{1152} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -46 a + 9\) , \( 135 a - 178\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-46a+9\right){x}+135a-178$ |
| 4356.6-b6 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{4} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.832988549$ |
$1.281706661$ |
3.228261005 |
\( \frac{56620795}{2304} a + \frac{115022599}{2304} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -33 a + 60\) , \( 45 a + 198\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-33a+60\right){x}+45a+198$ |
| 4356.6-b7 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{5} \cdot 3^{2} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$0.416494274$ |
$0.640853330$ |
3.228261005 |
\( -\frac{145011769343}{48} a + \frac{101553555457}{24} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -736 a + 159\) , \( 8283 a - 10030\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-736a+159\right){x}+8283a-10030$ |
| 4356.6-b8 |
4356.6-b |
$8$ |
$16$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{5} \cdot 3^{2} \cdot 11^{6} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1.665977099$ |
$0.640853330$ |
3.228261005 |
\( \frac{145011769343}{48} a + \frac{19365113857}{16} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -543 a + 960\) , \( 3123 a + 10482\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-543a+960\right){x}+3123a+10482$ |
| 4356.6-c1 |
4356.6-c |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 3^{4} \cdot 11^{9} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.737347964$ |
2.229530679 |
\( -\frac{11528611}{576} a - \frac{43599055}{288} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 128 a - 203\) , \( -919 a + 895\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(128a-203\right){x}-919a+895$ |
| 4356.6-c2 |
4356.6-c |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{7} \cdot 3^{8} \cdot 11^{9} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.737347964$ |
2.229530679 |
\( -\frac{45799}{1296} a + \frac{731363}{1296} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -61\) , \( -180 a + 113\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}-61{x}-180a+113$ |
| 4356.6-d1 |
4356.6-d |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{17} \cdot 3^{2} \cdot 11^{8} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.505633125$ |
2.293336294 |
\( -\frac{2553161206303}{1486848} a + \frac{335587165399}{495616} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -271 a - 454\) , \( -3958 a - 3029\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-271a-454\right){x}-3958a-3029$ |
| 4356.6-d2 |
4356.6-d |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 11^{14} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$0.252816562$ |
2.293336294 |
\( \frac{34147229879407}{61735357728} a + \frac{275344585935545}{277809109776} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( -257 a + 733\) , \( -4351 a + 2383\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-257a+733\right){x}-4351a+2383$ |
| 4356.6-d3 |
4356.6-d |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 11^{10} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \cdot 3 \) |
$1$ |
$0.505633125$ |
2.293336294 |
\( -\frac{102397297499}{134931456} a + \frac{150774304457}{67465728} \) |
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 103 a - 187\) , \( -463 a + 191\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(103a-187\right){x}-463a+191$ |
| 4356.6-d4 |
4356.6-d |
$4$ |
$4$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{23} \cdot 3^{2} \cdot 11^{8} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$0.505633125$ |
2.293336294 |
\( \frac{11336952210079}{380633088} a + \frac{71100053329}{63438848} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( 261 a - 12\) , \( -592 a - 2240\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(261a-12\right){x}-592a-2240$ |
| 4356.6-e1 |
4356.6-e |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{25} \cdot 3^{2} \cdot 11^{3} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3^{2} \cdot 7 \) |
$0.042930064$ |
$1.378554638$ |
5.636857313 |
\( -\frac{1217204443}{262144} a + \frac{470888579}{393216} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 2 a + 31\) , \( 46 a - 25\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(2a+31\right){x}+46a-25$ |
| 4356.6-e2 |
4356.6-e |
$2$ |
$2$ |
\(\Q(\sqrt{-7}) \) |
$2$ |
$[0, 1]$ |
4356.6 |
\( 2^{2} \cdot 3^{2} \cdot 11^{2} \) |
\( 2^{23} \cdot 3^{4} \cdot 11^{3} \) |
$1.92070$ |
$(a), (-a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3^{2} \cdot 7 \) |
$0.021465032$ |
$1.378554638$ |
5.636857313 |
\( \frac{20138209}{49152} a - \frac{111021713}{147456} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -7 a + 19\) , \( 43 a + 3\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7a+19\right){x}+43a+3$ |
*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.
