| 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 |
| 2.1-a1 |
2.1-a |
$4$ |
$6$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{3} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 3 \) |
$1$ |
$14.98573409$ |
2.05724399 |
\( \frac{990760012883838897}{8} a^{4} + \frac{620824910117852071}{8} a^{3} - \frac{1661559654070285375}{4} a^{2} - 347998563525778607 a - \frac{377189477685578601}{8} \) |
\( \bigl[a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -2 a^{4} + 5 a^{3} + 7 a^{2} - 14 a - 5\) , \( a^{3} - 4 a - 2\) , \( 191 a^{4} - 35 a^{3} - 961 a^{2} - 652 a - 105\) , \( 3832 a^{4} - 566 a^{3} - 19487 a^{2} - 13659 a - 1805\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(-2a^{4}+5a^{3}+7a^{2}-14a-5\right){x}^{2}+\left(191a^{4}-35a^{3}-961a^{2}-652a-105\right){x}+3832a^{4}-566a^{3}-19487a^{2}-13659a-1805$ |
| 2.1-a2 |
2.1-a |
$4$ |
$6$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{9} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$3641.533385$ |
2.05724399 |
\( \frac{28609729}{512} a^{4} + \frac{14983671}{512} a^{3} - \frac{48563491}{256} a^{2} - \frac{4370945}{32} a - \frac{3686985}{512} \) |
\( \bigl[a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -2 a^{4} + 5 a^{3} + 7 a^{2} - 14 a - 5\) , \( a^{3} - 4 a - 2\) , \( a^{4} + 15 a^{3} - 16 a^{2} - 72 a - 10\) , \( -21 a^{4} + 26 a^{3} + 91 a^{2} - 29 a - 8\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(-2a^{4}+5a^{3}+7a^{2}-14a-5\right){x}^{2}+\left(a^{4}+15a^{3}-16a^{2}-72a-10\right){x}-21a^{4}+26a^{3}+91a^{2}-29a-8$ |
| 2.1-a3 |
2.1-a |
$4$ |
$6$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{18} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1$ |
$1820.766692$ |
2.05724399 |
\( \frac{5323966132081}{262144} a^{4} - \frac{11934661930553}{262144} a^{3} - \frac{12273578458675}{131072} a^{2} + \frac{2379725624431}{16384} a + \frac{32206267289479}{262144} \) |
\( \bigl[a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -9 a^{4} + 26 a^{3} + a^{2} - 27 a - 6\) , \( 8 a^{4} - 55 a^{3} + 38 a^{2} + 70 a + 10\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-9a^{4}+26a^{3}+a^{2}-27a-6\right){x}+8a^{4}-55a^{3}+38a^{2}+70a+10$ |
| 2.1-a4 |
2.1-a |
$4$ |
$6$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{6} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$81$ |
\( 2 \cdot 3 \) |
$1$ |
$7.492867049$ |
2.05724399 |
\( \frac{12243490648960017213808305}{64} a^{4} - \frac{26584898010218540369944505}{64} a^{3} - \frac{28331071359310271740671763}{32} a^{2} + \frac{5198123563099020071039535}{4} a + \frac{71453295734956257649162567}{64} \) |
\( \bigl[a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -a^{3} + a^{2} + 4 a\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -334 a^{4} + 801 a^{3} + 46 a^{2} - 607 a - 121\) , \( -9137 a^{4} + 24278 a^{3} - 1664 a^{2} - 19882 a - 3260\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a\right){x}^{2}+\left(-334a^{4}+801a^{3}+46a^{2}-607a-121\right){x}-9137a^{4}+24278a^{3}-1664a^{2}-19882a-3260$ |
| 2.1-b1 |
2.1-b |
$2$ |
$3$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{24} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$0.311089713$ |
$2223.414605$ |
1.73670115 |
\( -\frac{140854061737}{16777216} a^{4} + \frac{521974831409}{16777216} a^{3} - \frac{86178293541}{8388608} a^{2} - \frac{34744410679}{1048576} a - \frac{76866520719}{16777216} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{2} + a + 2\) , \( -a^{4} + 3 a^{3} + 4 a^{2} - 9 a - 4\) , \( 4 a^{4} - 10 a^{3} - 19 a^{2} + 32 a + 27\) , \( 13 a^{4} - 30 a^{3} - 60 a^{2} + 95 a + 78\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{4}+3a^{3}+4a^{2}-9a-4\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(4a^{4}-10a^{3}-19a^{2}+32a+27\right){x}+13a^{4}-30a^{3}-60a^{2}+95a+78$ |
| 2.1-b2 |
2.1-b |
$2$ |
$3$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{72} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 2 \) |
$0.933269141$ |
$9.149854343$ |
1.73670115 |
\( \frac{10115169391867263832020327}{4722366482869645213696} a^{4} - \frac{36925691467527357603634015}{4722366482869645213696} a^{3} - \frac{15070876975854229243495765}{2361183241434822606848} a^{2} + \frac{7107199193045343615925177}{295147905179352825856} a + \frac{16131939070891235015385313}{4722366482869645213696} \) |
\( \bigl[a^{2} - a - 1\) , \( -a^{2} + a + 2\) , \( -a^{4} + 3 a^{3} + 4 a^{2} - 9 a - 4\) , \( -41 a^{4} + 90 a^{3} + 191 a^{2} - 288 a - 253\) , \( -432 a^{4} + 950 a^{3} + 1999 a^{2} - 3001 a - 2574\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(-a^{4}+3a^{3}+4a^{2}-9a-4\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(-41a^{4}+90a^{3}+191a^{2}-288a-253\right){x}-432a^{4}+950a^{3}+1999a^{2}-3001a-2574$ |
| 2.1-c1 |
2.1-c |
$6$ |
$8$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$0.194088284$ |
$702.0893015$ |
1.53965416 |
\( -\frac{6016314329794423}{2} a^{4} + \frac{16098709984645191}{2} a^{3} + 9604219291373887 a^{2} - 24555349614926227 a - \frac{8892891288053201}{2} \) |
\( \bigl[-a^{4} + 3 a^{3} + 3 a^{2} - 9 a - 3\) , \( a + 1\) , \( a\) , \( -187 a^{4} + 404 a^{3} + 865 a^{2} - 1257 a - 1082\) , \( 2434 a^{4} - 5293 a^{3} - 11263 a^{2} + 16576 a + 14230\bigr] \) |
${y}^2+\left(-a^{4}+3a^{3}+3a^{2}-9a-3\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-187a^{4}+404a^{3}+865a^{2}-1257a-1082\right){x}+2434a^{4}-5293a^{3}-11263a^{2}+16576a+14230$ |
| 2.1-c2 |
2.1-c |
$6$ |
$8$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{2} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.097044142$ |
$11233.42882$ |
1.53965416 |
\( -\frac{85802499}{4} a^{4} + \frac{236571483}{4} a^{3} + \frac{128113227}{2} a^{2} - 182270020 a - \frac{55692021}{4} \) |
\( \bigl[-a^{4} + 3 a^{3} + 3 a^{2} - 9 a - 3\) , \( a + 1\) , \( a\) , \( -197 a^{4} + 429 a^{3} + 910 a^{2} - 1342 a - 1147\) , \( 2173 a^{4} - 4718 a^{3} - 10057 a^{2} + 14760 a + 12682\bigr] \) |
${y}^2+\left(-a^{4}+3a^{3}+3a^{2}-9a-3\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-197a^{4}+429a^{3}+910a^{2}-1342a-1147\right){x}+2173a^{4}-4718a^{3}-10057a^{2}+14760a+12682$ |
| 2.1-c3 |
2.1-c |
$6$ |
$8$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2 \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.048522071$ |
$11233.42882$ |
1.53965416 |
\( \frac{9665}{2} a^{4} - \frac{13303}{2} a^{3} - 16437 a^{2} + 22088 a + \frac{18729}{2} \) |
\( \bigl[a^{3} - 3 a - 1\) , \( -2 a^{4} + 5 a^{3} + 8 a^{2} - 17 a - 9\) , \( -a^{4} + 3 a^{3} + 3 a^{2} - 8 a - 3\) , \( 5 a^{4} - 2 a^{3} - 19 a^{2} + 5 a + 11\) , \( 7 a^{4} + 7 a^{3} - 23 a^{2} - 28 a - 9\bigr] \) |
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(-a^{4}+3a^{3}+3a^{2}-8a-3\right){y}={x}^{3}+\left(-2a^{4}+5a^{3}+8a^{2}-17a-9\right){x}^{2}+\left(5a^{4}-2a^{3}-19a^{2}+5a+11\right){x}+7a^{4}+7a^{3}-23a^{2}-28a-9$ |
| 2.1-c4 |
2.1-c |
$6$ |
$8$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{8} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.388176569$ |
$702.0893015$ |
1.53965416 |
\( -\frac{20147727836671471}{256} a^{4} + \frac{1510035779603239}{256} a^{3} + \frac{51828624707413357}{128} a^{2} + \frac{4918069948271487}{16} a + \frac{10464859788169703}{256} \) |
\( \bigl[a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -a^{3} + 2 a^{2} + a - 3\) , \( -a^{4} + 3 a^{3} + 4 a^{2} - 9 a - 5\) , \( -40 a^{4} + 55 a^{3} + 138 a^{2} - 168 a - 31\) , \( -532 a^{4} + 1010 a^{3} + 1718 a^{2} - 2968 a - 552\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}{y}+\left(-a^{4}+3a^{3}+4a^{2}-9a-5\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+a-3\right){x}^{2}+\left(-40a^{4}+55a^{3}+138a^{2}-168a-31\right){x}-532a^{4}+1010a^{3}+1718a^{2}-2968a-552$ |
| 2.1-c5 |
2.1-c |
$6$ |
$8$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{4} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.194088284$ |
$5616.714412$ |
1.53965416 |
\( \frac{1253080213679809}{16} a^{4} - \frac{2720875245575881}{16} a^{3} - \frac{2899590144962547}{8} a^{2} + 532010541528938 a + \frac{7313005461632695}{16} \) |
\( \bigl[a^{2} - 2\) , \( a^{2} - 1\) , \( a^{3} - 4 a - 2\) , \( -3 a^{4} + 12 a^{3} + 15 a^{2} - 42 a - 35\) , \( 11 a^{4} - 20 a^{3} - 51 a^{2} + 68 a + 64\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-4a-2\right){y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(-3a^{4}+12a^{3}+15a^{2}-42a-35\right){x}+11a^{4}-20a^{3}-51a^{2}+68a+64$ |
| 2.1-c6 |
2.1-c |
$6$ |
$8$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
2.1 |
\( 2 \) |
\( 2^{2} \) |
$42.38186$ |
$(a^4-2a^3-5a^2+7a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.388176569$ |
$702.0893015$ |
1.53965416 |
\( \frac{111581902846313479640990352771}{4} a^{4} - \frac{242283315437279062404943315007}{4} a^{3} - \frac{258197187598215990915659337479}{2} a^{2} + 189493841268254658660383521177 a + \frac{651194576068388032235308058749}{4} \) |
\( \bigl[a^{3} - 4 a - 2\) , \( 2 a^{4} - 5 a^{3} - 7 a^{2} + 14 a + 5\) , \( -a^{4} + 3 a^{3} + 3 a^{2} - 8 a - 3\) , \( 7 a^{4} + 45 a^{3} - 116 a^{2} - 134 a - 18\) , \( -101 a^{4} - 232 a^{3} + 929 a^{2} + 895 a + 125\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(-a^{4}+3a^{3}+3a^{2}-8a-3\right){y}={x}^{3}+\left(2a^{4}-5a^{3}-7a^{2}+14a+5\right){x}^{2}+\left(7a^{4}+45a^{3}-116a^{2}-134a-18\right){x}-101a^{4}-232a^{3}+929a^{2}+895a+125$ |
*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.
