| 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 |
| 11.1-a1 |
11.1-a |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$4290.634146$ |
2.42394628 |
\( -\frac{2769926}{11} a^{4} - \frac{7424323}{11} a^{3} + \frac{16111424}{11} a^{2} + 4521940 a + \frac{24581940}{11} \) |
\( \bigl[1\) , \( -a^{3} + 2 a^{2} + a - 3\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 3\) , \( a^{3} - 4 a + 1\) , \( -2 a^{3} + 2 a^{2} + 4 a - 1\bigr] \) |
${y}^2+{x}{y}+\left(a^{4}-2a^{3}-4a^{2}+6a+3\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+a-3\right){x}^{2}+\left(a^{3}-4a+1\right){x}-2a^{3}+2a^{2}+4a-1$ |
| 11.1-a2 |
11.1-a |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{4} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1072.658536$ |
2.42394628 |
\( \frac{6322331268044115}{14641} a^{4} - \frac{23418956268286453}{14641} a^{3} + \frac{8298211583434928}{14641} a^{2} + \frac{2162946184419235}{1331} a + \frac{3709886035859698}{14641} \) |
\( \bigl[a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( -a^{4} + 2 a^{3} + 4 a^{2} - 6 a - 2\) , \( -a^{4} + 3 a^{3} + 3 a^{2} - 9 a - 2\) , \( -121 a^{4} + 251 a^{3} + 547 a^{2} - 779 a - 654\) , \( 938 a^{4} - 2196 a^{3} - 4415 a^{2} + 6919 a + 5833\bigr] \) |
${y}^2+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){x}{y}+\left(-a^{4}+3a^{3}+3a^{2}-9a-2\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+4a^{2}-6a-2\right){x}^{2}+\left(-121a^{4}+251a^{3}+547a^{2}-779a-654\right){x}+938a^{4}-2196a^{3}-4415a^{2}+6919a+5833$ |
| 11.1-a3 |
11.1-a |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{2} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$8581.268293$ |
2.42394628 |
\( \frac{60288096232545}{121} a^{4} - \frac{130906555597021}{121} a^{3} - \frac{279009609994653}{121} a^{2} + \frac{37230610788498}{11} a + \frac{351842646024300}{121} \) |
\( \bigl[a^{2} - 2\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 8 a - 4\) , \( -a^{4} + 3 a^{3} + 4 a^{2} - 9 a - 4\) , \( 2 a^{4} - a^{3} - 9 a^{2} - 9 a - 3\) , \( 6 a^{4} - 7 a^{3} - 26 a^{2} + 4 a\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{4}+3a^{3}+4a^{2}-9a-4\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+5a^{2}-8a-4\right){x}^{2}+\left(2a^{4}-a^{3}-9a^{2}-9a-3\right){x}+6a^{4}-7a^{3}-26a^{2}+4a$ |
| 11.1-a4 |
11.1-a |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11 \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$4290.634146$ |
2.42394628 |
\( \frac{12419405910755679760810715}{11} a^{4} - \frac{26966871536181553304111845}{11} a^{3} - \frac{57476268032719258975075064}{11} a^{2} + 7669542599378813017784559 a + \frac{72479941287745983554396814}{11} \) |
\( \bigl[a^{2} - 2\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 8 a - 4\) , \( -a^{4} + 3 a^{3} + 4 a^{2} - 9 a - 4\) , \( 22 a^{4} - 26 a^{3} - 94 a^{2} + 21 a + 2\) , \( -83 a^{4} + 47 a^{3} + 395 a^{2} + 133 a + 13\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(-a^{4}+3a^{3}+4a^{2}-9a-4\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+5a^{2}-8a-4\right){x}^{2}+\left(22a^{4}-26a^{3}-94a^{2}+21a+2\right){x}-83a^{4}+47a^{3}+395a^{2}+133a+13$ |
| 11.1-b1 |
11.1-b |
$2$ |
$5$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( -11 \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$625$ |
\( 1 \) |
$1$ |
$0.474535899$ |
0.670209512 |
\( -\frac{12430061819623358548999646912000016}{11} a^{4} + \frac{46043025029690426679281077283115642}{11} a^{3} - \frac{16314695587693936002344398181903027}{11} a^{2} - 4252491343722498186300498152425006 a - \frac{7293922300434311569558626398695053}{11} \) |
\( \bigl[-a^{4} + 3 a^{3} + 3 a^{2} - 8 a - 3\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 5 a + 1\) , \( a^{3} - a^{2} - 3 a\) , \( -86 a^{4} + 3 a^{3} + 193 a^{2} + 199 a + 60\) , \( -4962 a^{4} - 2085 a^{3} + 15827 a^{2} + 12029 a + 1283\bigr] \) |
${y}^2+\left(-a^{4}+3a^{3}+3a^{2}-8a-3\right){x}{y}+\left(a^{3}-a^{2}-3a\right){y}={x}^{3}+\left(a^{4}-2a^{3}-3a^{2}+5a+1\right){x}^{2}+\left(-86a^{4}+3a^{3}+193a^{2}+199a+60\right){x}-4962a^{4}-2085a^{3}+15827a^{2}+12029a+1283$ |
| 11.1-b2 |
11.1-b |
$2$ |
$5$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{5} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$1482.924685$ |
0.670209512 |
\( -\frac{51581336262}{161051} a^{4} + \frac{189231965379}{161051} a^{3} - \frac{60831697345}{161051} a^{2} - \frac{17940053555}{14641} a - \frac{36167170459}{161051} \) |
\( \bigl[a^{2} - a - 2\) , \( -a^{3} + 3 a + 2\) , \( a^{4} - 2 a^{3} - 3 a^{2} + 6 a + 1\) , \( -a^{4} + 4 a^{2} + 9 a + 3\) , \( -a^{4} + 2 a^{3} - 3 a^{2} + 3 a + 1\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{4}-2a^{3}-3a^{2}+6a+1\right){y}={x}^{3}+\left(-a^{3}+3a+2\right){x}^{2}+\left(-a^{4}+4a^{2}+9a+3\right){x}-a^{4}+2a^{3}-3a^{2}+3a+1$ |
| 11.1-c1 |
11.1-c |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{28} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 7 \) |
$1$ |
$44.34525002$ |
0.701466031 |
\( -\frac{2814511163752348580201349601995551}{144209936106499234037676064081} a^{4} + \frac{7024156379416378787315857585615865}{144209936106499234037676064081} a^{3} + \frac{13793385475428462351729701616182427}{144209936106499234037676064081} a^{2} - \frac{2263342866356768211691854556627463}{13109994191499930367061460371} a - \frac{20152664762690240711849264417137559}{144209936106499234037676064081} \) |
\( \bigl[1\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 3\) , \( a^{2} - a - 1\) , \( -242 a^{4} + 903 a^{3} - 350 a^{2} - 875 a - 145\) , \( 7309 a^{4} - 27093 a^{3} + 9670 a^{2} + 27452 a + 4253\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-1\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+6a+3\right){x}^{2}+\left(-242a^{4}+903a^{3}-350a^{2}-875a-145\right){x}+7309a^{4}-27093a^{3}+9670a^{2}+27452a+4253$ |
| 11.1-c2 |
11.1-c |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{14} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \cdot 7 \) |
$1$ |
$354.7620001$ |
0.701466031 |
\( \frac{944469806939641457726850}{379749833583241} a^{4} - \frac{2122415087246029627252295}{379749833583241} a^{3} - \frac{4343198443816886016084325}{379749833583241} a^{2} + \frac{618687637239948673891676}{34522712143931} a + \frac{5741045574304169608952431}{379749833583241} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 1\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 4\) , \( -a^{4} + 3 a^{3} + 3 a^{2} - 8 a - 3\) , \( -1441 a^{4} + 3138 a^{3} + 6662 a^{2} - 9819 a - 8428\) , \( -46680 a^{4} + 101352 a^{3} + 216018 a^{2} - 317028 a - 272375\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+1\right){x}{y}+\left(-a^{4}+3a^{3}+3a^{2}-8a-3\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+6a+4\right){x}^{2}+\left(-1441a^{4}+3138a^{3}+6662a^{2}-9819a-8428\right){x}-46680a^{4}+101352a^{3}+216018a^{2}-317028a-272375$ |
| 11.1-c3 |
11.1-c |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{7} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 7 \) |
$1$ |
$177.3810000$ |
0.701466031 |
\( -\frac{3069670424755009349061343}{19487171} a^{4} + \frac{299616585060440803764625}{19487171} a^{3} + \frac{15970547202651724095494599}{19487171} a^{2} + \frac{1099775811878439791091717}{1771561} a + \frac{1608154668840005272040881}{19487171} \) |
\( \bigl[a^{3} - a^{2} - 2 a + 1\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 6 a + 4\) , \( -a^{4} + 3 a^{3} + 3 a^{2} - 8 a - 3\) , \( -1441 a^{4} + 3168 a^{3} + 6522 a^{2} - 9934 a - 8458\) , \( -46859 a^{4} + 101626 a^{3} + 216356 a^{2} - 315240 a - 271139\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-2a+1\right){x}{y}+\left(-a^{4}+3a^{3}+3a^{2}-8a-3\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+6a+4\right){x}^{2}+\left(-1441a^{4}+3168a^{3}+6522a^{2}-9934a-8458\right){x}-46859a^{4}+101626a^{3}+216356a^{2}-315240a-271139$ |
| 11.1-c4 |
11.1-c |
$4$ |
$4$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( - 11^{7} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 7 \) |
$1$ |
$177.3810000$ |
0.701466031 |
\( \frac{590732269314467338033745094}{19487171} a^{4} - \frac{1282686251931268648314475520}{19487171} a^{3} - \frac{2733873624123208154740134995}{19487171} a^{2} + \frac{364803786661913751909779555}{1771561} a + \frac{3447527241187039976786661971}{19487171} \) |
\( \bigl[a^{3} - 4 a - 2\) , \( -a^{3} + 2 a^{2} + 2 a - 3\) , \( a^{2} - 2\) , \( -28 a^{4} + 53 a^{3} + 139 a^{2} - 170 a - 177\) , \( -157 a^{4} + 362 a^{3} + 672 a^{2} - 1126 a - 750\bigr] \) |
${y}^2+\left(a^{3}-4a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+2a-3\right){x}^{2}+\left(-28a^{4}+53a^{3}+139a^{2}-170a-177\right){x}-157a^{4}+362a^{3}+672a^{2}-1126a-750$ |
| 11.1-d1 |
11.1-d |
$2$ |
$5$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11^{5} \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$1769.458286$ |
0.799708701 |
\( \frac{106476089}{161051} a^{4} - \frac{198113042}{161051} a^{3} - \frac{361568756}{161051} a^{2} + \frac{44783789}{14641} a + \frac{9814598}{161051} \) |
\( \bigl[a^{2} - 1\) , \( a^{3} - a^{2} - 4 a\) , \( -a^{4} + 3 a^{3} + 3 a^{2} - 9 a - 3\) , \( -81 a^{4} + 178 a^{3} + 376 a^{2} - 557 a - 475\) , \( -1268 a^{4} + 2756 a^{3} + 5870 a^{2} - 8623 a - 7409\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(-a^{4}+3a^{3}+3a^{2}-9a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a\right){x}^{2}+\left(-81a^{4}+178a^{3}+376a^{2}-557a-475\right){x}-1268a^{4}+2756a^{3}+5870a^{2}-8623a-7409$ |
| 11.1-d2 |
11.1-d |
$2$ |
$5$ |
5.5.195829.1 |
$5$ |
$[5, 0]$ |
11.1 |
\( 11 \) |
\( 11 \) |
$50.25929$ |
$(a^4-2a^3-4a^2+6a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$625$ |
\( 1 \) |
$1$ |
$0.566226651$ |
0.799708701 |
\( \frac{826166701116865630}{11} a^{4} - \frac{2213393456654290609}{11} a^{3} - \frac{2627862618618531304}{11} a^{2} + 613135222427199995 a + \frac{1202837986429448866}{11} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + 4 a + 2\) , \( -a^{4} + 3 a^{3} + 4 a^{2} - 10 a - 5\) , \( -57 a^{4} + 273 a^{3} - 207 a^{2} - 340 a - 51\) , \( -812 a^{4} + 5685 a^{3} - 5890 a^{2} - 8747 a - 1304\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(-a^{4}+3a^{3}+4a^{2}-10a-5\right){y}={x}^{3}+\left(-a^{3}+4a+2\right){x}^{2}+\left(-57a^{4}+273a^{3}-207a^{2}-340a-51\right){x}-812a^{4}+5685a^{3}-5890a^{2}-8747a-1304$ |
*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.
