| 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 |
| 5200.6-a1 |
5200.6-a |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{12} \cdot 5^{3} \cdot 13^{3} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.073910902$ |
$2.435961901$ |
2.160529725 |
\( \frac{906876}{2197} a + \frac{1118799}{2197} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( -i + 5\) , \( 8 i - 5\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-i+5\right){x}+8i-5$ |
| 5200.6-a2 |
5200.6-a |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{12} \cdot 5^{3} \cdot 13^{6} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.147821805$ |
$1.217980950$ |
2.160529725 |
\( -\frac{10047446145}{4826809} a + \frac{17756992962}{4826809} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( 19 i - 35\) , \( 44 i - 57\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(19i-35\right){x}+44i-57$ |
| 5200.6-b1 |
5200.6-b |
$6$ |
$18$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{30} \cdot 5^{15} \cdot 13 \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$1.318760694$ |
$0.214824951$ |
2.266421617 |
\( -\frac{276861163011391}{13000000000} a - \frac{33515586556057}{812500000} \) |
\( \bigl[i + 1\) , \( -i - 1\) , \( i + 1\) , \( -269 i - 2036\) , \( 7244 i + 36407\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-269i-2036\right){x}+7244i+36407$ |
| 5200.6-b2 |
5200.6-b |
$6$ |
$18$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{18} \cdot 5^{9} \cdot 13^{3} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \) |
$0.439586898$ |
$0.644474854$ |
2.266421617 |
\( \frac{37525044319}{2197000} a - \frac{7169596274}{274625} \) |
\( \bigl[i + 1\) , \( -i - 1\) , \( i + 1\) , \( -189 i - 96\) , \( -1128 i + 111\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-189i-96\right){x}-1128i+111$ |
| 5200.6-b3 |
5200.6-b |
$6$ |
$18$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{15} \cdot 5^{12} \cdot 13^{6} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Cs |
$1$ |
\( 2^{4} \) |
$0.879173796$ |
$0.322237427$ |
2.266421617 |
\( \frac{133816114442969}{301675562500} a - \frac{19082395919017}{301675562500} \) |
\( \bigl[i + 1\) , \( -i - 1\) , \( i + 1\) , \( -229 i + 184\) , \( -2832 i - 2761\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(-229i+184\right){x}-2832i-2761$ |
| 5200.6-b4 |
5200.6-b |
$6$ |
$18$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{21} \cdot 5^{24} \cdot 13^{2} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$2.637521388$ |
$0.107412475$ |
2.266421617 |
\( -\frac{8418015312387897223}{20629882812500000} a + \frac{2783266907131437289}{20629882812500000} \) |
\( \bigl[i + 1\) , \( -i - 1\) , \( i + 1\) , \( 1971 i - 1716\) , \( 86412 i + 51831\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(1971i-1716\right){x}+86412i+51831$ |
| 5200.6-b5 |
5200.6-b |
$6$ |
$18$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{14} \cdot 5^{7} \cdot 13 \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.146528966$ |
$1.933424563$ |
2.266421617 |
\( \frac{31409}{130} a + \frac{101344}{65} \) |
\( \bigl[i + 1\) , \( -i - 1\) , \( i + 1\) , \( 11 i + 4\) , \( -12 i - 1\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(11i+4\right){x}-12i-1$ |
| 5200.6-b6 |
5200.6-b |
$6$ |
$18$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{13} \cdot 5^{8} \cdot 13^{2} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \) |
$0.293057932$ |
$0.966712281$ |
2.266421617 |
\( \frac{4406742137}{8450} a + \frac{1310300809}{8450} \) |
\( \bigl[i + 1\) , \( -i - 1\) , \( i + 1\) , \( 151 i + 24\) , \( -440 i - 605\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i-1\right){x}^{2}+\left(151i+24\right){x}-440i-605$ |
| 5200.6-c1 |
5200.6-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{8} \cdot 5^{8} \cdot 13^{2} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.699559292$ |
1.699559292 |
\( -\frac{8646624}{4225} a - \frac{66027268}{4225} \) |
\( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( -20 i + 18\) , \( -4 i - 48\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-20i+18\right){x}-4i-48$ |
| 5200.6-c2 |
5200.6-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{11} \cdot 5^{8} \cdot 13^{8} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.424889823$ |
1.699559292 |
\( \frac{94078100761841}{20393268025} a - \frac{11284537597913}{20393268025} \) |
\( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( 260 i - 192\) , \( -2412 i + 708\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(260i-192\right){x}-2412i+708$ |
| 5200.6-c3 |
5200.6-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{10} \cdot 5^{10} \cdot 13^{4} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.849779646$ |
1.699559292 |
\( -\frac{15461171586}{17850625} a - \frac{9080741152}{17850625} \) |
\( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( -50 i - 22\) , \( -224 i - 8\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-50i-22\right){x}-224i-8$ |
| 5200.6-c4 |
5200.6-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{4} \cdot 5^{7} \cdot 13 \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$3.399118585$ |
1.699559292 |
\( \frac{75008}{65} a - \frac{29104}{65} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( i + 3\) , \( -i + 3\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(i+1\right){x}^{2}+\left(i+3\right){x}-i+3$ |
| 5200.6-c5 |
5200.6-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{11} \cdot 5^{14} \cdot 13^{2} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.424889823$ |
1.699559292 |
\( \frac{143087370512191}{66015625} a + \frac{29292377558137}{66015625} \) |
\( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( -840 i - 492\) , \( -11636 i + 476\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-840i-492\right){x}-11636i+476$ |
| 5200.6-c6 |
5200.6-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{10} \cdot 5^{7} \cdot 13 \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.849779646$ |
1.699559292 |
\( \frac{4355686402}{65} a + \frac{17124606704}{65} \) |
\( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( -310 i + 298\) , \( -976 i - 3144\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-310i+298\right){x}-976i-3144$ |
| 5200.6-d1 |
5200.6-d |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{8} \cdot 5^{9} \cdot 13 \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.119999649$ |
2.119999649 |
\( \frac{2224}{13} a + \frac{356}{13} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -5\) , \( -8 i - 12\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}-5{x}-8i-12$ |
| 5200.6-d2 |
5200.6-d |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5200.6 |
\( 2^{4} \cdot 5^{2} \cdot 13 \) |
\( 2^{10} \cdot 5^{9} \cdot 13^{2} \) |
$1.51764$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.059999824$ |
2.119999649 |
\( -\frac{47776420}{169} a + \frac{17266442}{169} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -20 i - 115\) , \( -136 i - 466\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-20i-115\right){x}-136i-466$ |
*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.
