| 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 |
| 27225.8-a1 |
27225.8-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 5^{9} \cdot 11^{3} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.502834080$ |
$0.750726747$ |
3.642170149 |
\( \frac{24171483673}{390625} a - \frac{40464261007}{390625} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 119 a - 96\) , \( -562 a - 402\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(119a-96\right){x}-562a-402$ |
| 27225.8-a2 |
27225.8-a |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 5^{6} \cdot 11^{3} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.251417040$ |
$1.501453495$ |
3.642170149 |
\( \frac{13891}{625} a + \frac{147806}{625} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( 4 a - 11\) , \( -13 a - 33\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-11\right){x}-13a-33$ |
| 27225.8-b1 |
27225.8-b |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.232902986$ |
1.486936948 |
\( \frac{59319}{55} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 9 a + 19\) , \( 10 a + 20\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(9a+19\right){x}+10a+20$ |
| 27225.8-b2 |
27225.8-b |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{4} \cdot 11^{10} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.616451493$ |
1.486936948 |
\( \frac{8120601}{3025} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -46 a - 91\) , \( 219 a + 141\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-46a-91\right){x}+219a+141$ |
| 27225.8-b3 |
27225.8-b |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{2} \cdot 11^{14} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.308225746$ |
1.486936948 |
\( \frac{2749884201}{73205} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -321 a - 641\) , \( -4896 a - 4644\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-321a-641\right){x}-4896a-4644$ |
| 27225.8-b4 |
27225.8-b |
$4$ |
$4$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{8} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.308225746$ |
1.486936948 |
\( \frac{22930509321}{6875} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -651 a - 1301\) , \( 16070 a + 13330\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-651a-1301\right){x}+16070a+13330$ |
| 27225.8-c1 |
27225.8-c |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{11} \cdot 5^{6} \cdot 11^{3} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.031866042$ |
4.977909085 |
\( -\frac{2797353512}{151875} a - \frac{325474339}{50625} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -48 a + 53\) , \( -37 a + 301\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-48a+53\right){x}-37a+301$ |
| 27225.8-c2 |
27225.8-c |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{16} \cdot 5^{9} \cdot 11^{3} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.515933021$ |
4.977909085 |
\( \frac{17874017121802}{23066015625} a + \frac{4304613879119}{7688671875} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -58 a + 168\) , \( -458 a - 55\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-58a+168\right){x}-458a-55$ |
| 27225.8-d1 |
27225.8-d |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{4} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.2 |
$4$ |
\( 1 \) |
$1$ |
$0.897405682$ |
2.164623951 |
\( \frac{206103}{125} a + \frac{264299}{125} \) |
\( \bigl[1\) , \( -a\) , \( 1\) , \( -42 a + 7\) , \( 98 a - 218\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-42a+7\right){x}+98a-218$ |
| 27225.8-d2 |
27225.8-d |
$2$ |
$3$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{4} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3B.1.1 |
$4$ |
\( 3^{2} \) |
$1$ |
$0.897405682$ |
2.164623951 |
\( -\frac{206103}{125} a + \frac{470402}{125} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 8 a - 73\) , \( -58 a + 217\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(8a-73\right){x}-58a+217$ |
| 27225.8-e1 |
27225.8-e |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{9} \cdot 11^{9} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$0.392054257$ |
1.891340900 |
\( \frac{24171483673}{390625} a - \frac{40464261007}{390625} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 410 a + 85\) , \( 1193 a + 6803\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(410a+85\right){x}+1193a+6803$ |
| 27225.8-e2 |
27225.8-e |
$2$ |
$2$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{6} \cdot 11^{9} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.784108514$ |
1.891340900 |
\( \frac{13891}{625} a + \frac{147806}{625} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( 25 a - 25\) , \( -50 a + 225\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(25a-25\right){x}-50a+225$ |
| 27225.8-f1 |
27225.8-f |
$4$ |
$6$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 5^{16} \cdot 11^{9} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$1$ |
$0.116885974$ |
3.383274952 |
\( \frac{1006933623697}{29541015625} a + \frac{46653574001501}{29541015625} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -1739 a - 885\) , \( -2648 a + 24541\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-1739a-885\right){x}-2648a+24541$ |
| 27225.8-f2 |
27225.8-f |
$4$ |
$6$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{26} \cdot 11^{8} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{7} \cdot 3 \) |
$1$ |
$0.058442987$ |
3.383274952 |
\( -\frac{2638659751550362590877}{655651092529296875} a + \frac{638610383092162643063}{59604644775390625} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 3646 a - 21896\) , \( 302918 a - 1131650\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3646a-21896\right){x}+302918a-1131650$ |
| 27225.8-f3 |
27225.8-f |
$4$ |
$6$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{9} \cdot 5^{14} \cdot 11^{12} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{7} \cdot 3 \) |
$1$ |
$0.058442987$ |
3.383274952 |
\( \frac{87281240121023}{519921875} a + \frac{57342994339168}{519921875} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -21649 a - 4570\) , \( -1749954 a + 1418945\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-21649a-4570\right){x}-1749954a+1418945$ |
| 27225.8-f4 |
27225.8-f |
$4$ |
$6$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{3} \cdot 5^{16} \cdot 11^{7} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{6} \cdot 3 \) |
$1$ |
$0.116885974$ |
3.383274952 |
\( -\frac{172021249885174171}{2685546875} a + \frac{135400285739660314}{2685546875} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 3756 a - 21511\) , \( 295262 a - 1181546\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(3756a-21511\right){x}+295262a-1181546$ |
| 27225.8-g1 |
27225.8-g |
$1$ |
$1$ |
\(\Q(\sqrt{-11}) \) |
$2$ |
$[0, 1]$ |
27225.8 |
\( 3^{2} \cdot 5^{2} \cdot 11^{2} \) |
\( 3^{6} \cdot 5^{10} \cdot 11^{10} \) |
$3.80695$ |
$(a-1), (-a-1), (a-2), (-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
|
|
$4$ |
\( 5 \) |
$1$ |
$0.281794328$ |
3.398567471 |
\( -\frac{1459161}{3125} \) |
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -129 a - 256\) , \( 3062 a + 2536\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-129a-256\right){x}+3062a+2536$ |
*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.
