| 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 |
| 12.1-a1 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{18} \cdot 3^{2} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$31.38748448$ |
1.477069450 |
\( \frac{767765770888643}{36864} a^{2} + \frac{356193608746913}{9216} a - \frac{268832552819119}{18432} \) |
\( \bigl[a^{2} - a - 3\) , \( 1\) , \( a + 1\) , \( 57167 a^{2} - 38782 a - 355472\) , \( -13151176 a^{2} + 8921268 a + 81776467\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(57167a^{2}-38782a-355472\right){x}-13151176a^{2}+8921268a+81776467$ |
| 12.1-a2 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{9} \cdot 3^{4} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$3.923435560$ |
1.477069450 |
\( \frac{28680482699892979305475847}{5184} a^{2} + \frac{13306112817877625586780929}{1296} a - \frac{10042985788695646918388803}{2592} \) |
\( \bigl[a^{2} - a - 3\) , \( 1\) , \( a + 1\) , \( 39197 a^{2} - 26592 a - 243732\) , \( -21633056 a^{2} + 14675060 a + 134518379\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(39197a^{2}-26592a-243732\right){x}-21633056a^{2}+14675060a+134518379$ |
| 12.1-a3 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{18} \cdot 3 \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$31.38748448$ |
1.477069450 |
\( -\frac{4929322714393}{96} a^{2} + \frac{1671929742829}{48} a + \frac{61302855066215}{192} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 5\) , \( 1\) , \( 7982841864 a^{2} - 5415263165 a - 49638801124\) , \( -690834441982159 a^{2} + 468636404179256 a + 4295737540690083\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(7982841864a^{2}-5415263165a-49638801124\right){x}-690834441982159a^{2}+468636404179256a+4295737540690083$ |
| 12.1-a4 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{27} \cdot 3 \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$31.38748448$ |
1.477069450 |
\( \frac{19526696771202314825}{50331648} a^{2} - \frac{15511076285266340125}{12582912} a + \frac{8967854540137295603}{25165824} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 5\) , \( 1\) , \( -109 a^{2} + 330 a - 78\) , \( 1316 a^{2} - 4197 a + 1221\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(-109a^{2}+330a-78\right){x}+1316a^{2}-4197a+1221$ |
| 12.1-a5 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{6} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$94.16245344$ |
1.477069450 |
\( -\frac{5004517}{11664} a^{2} + \frac{2632469}{2916} a + \frac{47382317}{5832} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -5635 a^{2} - 10459 a + 3945\) , \( -510983 a^{2} - 948268 a + 357859\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5635a^{2}-10459a+3945\right){x}-510983a^{2}-948268a+357859$ |
| 12.1-a6 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{3} \cdot 3^{12} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 2^{2} \) |
$1$ |
$11.77030668$ |
1.477069450 |
\( \frac{8749450000313}{2125764} a^{2} + \frac{3432109524343}{1062882} a - \frac{1556232857809}{1062882} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -86920 a^{2} - 161304 a + 60875\) , \( -35286317 a^{2} - 65483378 a + 24712271\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-86920a^{2}-161304a+60875\right){x}-35286317a^{2}-65483378a+24712271$ |
| 12.1-a7 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{3} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \) |
$1$ |
$94.16245344$ |
1.477069450 |
\( -\frac{61321}{54} a^{2} + \frac{18883}{27} a + \frac{814559}{108} \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( 8158 a^{2} - 5530 a - 50714\) , \( -618317 a^{2} + 419416 a + 3844749\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(8158a^{2}-5530a-50714\right){x}-618317a^{2}+419416a+3844749$ |
| 12.1-a8 |
12.1-a |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{9} \cdot 3^{3} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$1$ |
$94.16245344$ |
1.477069450 |
\( -\frac{63031725775}{6912} a^{2} + \frac{10686570875}{1728} a + \frac{196019581547}{3456} \) |
\( \bigl[1\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( -6522922 a^{2} - 12105060 a + 4568240\) , \( 20564626474 a^{2} + 38163268421 a - 14402146127\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-6522922a^{2}-12105060a+4568240\right){x}+20564626474a^{2}+38163268421a-14402146127$ |
| 12.1-b1 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{18} \cdot 3^{2} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$1$ |
$6.400037789$ |
0.903541694 |
\( \frac{767765770888643}{36864} a^{2} + \frac{356193608746913}{9216} a - \frac{268832552819119}{18432} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 5\) , \( a^{2} - a - 4\) , \( 333088755 a^{2} - 225955030 a - 2071208065\) , \( 5879094630820 a^{2} - 3988159246537 a - 36557308055487\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(333088755a^{2}-225955030a-2071208065\right){x}+5879094630820a^{2}-3988159246537a-36557308055487$ |
| 12.1-b2 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{9} \cdot 3^{4} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{3} \) |
$1$ |
$1.600009447$ |
0.903541694 |
\( \frac{28680482699892979305475847}{5184} a^{2} + \frac{13306112817877625586780929}{1296} a - \frac{10042985788695646918388803}{2592} \) |
\( \bigl[a^{2} - a - 3\) , \( a^{2} - a - 5\) , \( a^{2} - a - 4\) , \( 228386865 a^{2} - 154929160 a - 1420152165\) , \( 9641929510302 a^{2} - 6540726548163 a - 59955317866391\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-5\right){x}^{2}+\left(228386865a^{2}-154929160a-1420152165\right){x}+9641929510302a^{2}-6540726548163a-59955317866391$ |
| 12.1-b3 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{18} \cdot 3 \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2^{2} \) |
$1$ |
$3.200018894$ |
0.903541694 |
\( -\frac{4929322714393}{96} a^{2} + \frac{1671929742829}{48} a + \frac{61302855066215}{192} \) |
\( \bigl[a^{2} - a - 3\) , \( 0\) , \( 1\) , \( 46512244972505 a^{2} - 31552177930907 a - 289221823186578\) , \( 307256602036167686328 a^{2} - 208431456783512664924 a - 1910578917004437407558\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+{y}={x}^{3}+\left(46512244972505a^{2}-31552177930907a-289221823186578\right){x}+307256602036167686328a^{2}-208431456783512664924a-1910578917004437407558$ |
| 12.1-b4 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{27} \cdot 3 \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$6.400037789$ |
0.903541694 |
\( \frac{19526696771202314825}{50331648} a^{2} - \frac{15511076285266340125}{12582912} a + \frac{8967854540137295603}{25165824} \) |
\( \bigl[a^{2} - a - 3\) , \( 0\) , \( 1\) , \( 3100 a^{2} - 1980 a - 19628\) , \( 36343 a^{2} - 23309 a - 229828\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+{y}={x}^{3}+\left(3100a^{2}-1980a-19628\right){x}+36343a^{2}-23309a-229828$ |
| 12.1-b5 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{6} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$172.8010203$ |
0.903541694 |
\( -\frac{5004517}{11664} a^{2} + \frac{2632469}{2916} a + \frac{47382317}{5832} \) |
\( \bigl[1\) , \( a^{2} - 4\) , \( a + 1\) , \( 1385 a^{2} - 944 a - 8621\) , \( -46274 a^{2} + 31374 a + 287704\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(1385a^{2}-944a-8621\right){x}-46274a^{2}+31374a+287704$ |
| 12.1-b6 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( - 2^{3} \cdot 3^{12} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$43.20025507$ |
0.903541694 |
\( \frac{8749450000313}{2125764} a^{2} + \frac{3432109524343}{1062882} a - \frac{1556232857809}{1062882} \) |
\( \bigl[1\) , \( a^{2} - 4\) , \( a + 1\) , \( -2020 a^{2} + 1291 a + 12389\) , \( -225678 a^{2} + 152356 a + 1401706\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-2020a^{2}+1291a+12389\right){x}-225678a^{2}+152356a+1401706$ |
| 12.1-b7 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{6} \cdot 3^{3} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$86.40051015$ |
0.903541694 |
\( -\frac{61321}{54} a^{2} + \frac{18883}{27} a + \frac{814559}{108} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( 47521831 a^{2} - 32237043 a - 295499615\) , \( 279209071496 a^{2} - 189405054711 a - 1736174135565\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(47521831a^{2}-32237043a-295499615\right){x}+279209071496a^{2}-189405054711a-1736174135565$ |
| 12.1-b8 |
12.1-b |
$8$ |
$12$ |
3.3.1016.1 |
$3$ |
$[3, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{9} \cdot 3^{3} \) |
$4.30972$ |
$(-a), (-a+3), (a^2-a-5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$172.8010203$ |
0.903541694 |
\( -\frac{63031725775}{6912} a^{2} + \frac{10686570875}{1728} a + \frac{196019581547}{3456} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( -2369 a^{2} - 4397 a + 1659\) , \( 139103 a^{2} + 258144 a - 97419\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-2369a^{2}-4397a+1659\right){x}+139103a^{2}+258144a-97419$ |
*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.
