| 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 |
| 8.1-a1 |
8.1-a |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{43} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$4.011156839$ |
1.492523475 |
\( \frac{3627718341771149}{8589934592} a^{2} + \frac{12708628339978831}{8589934592} a + \frac{3221718432139487}{4294967296} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 6\) , \( 0\) , \( \frac{15}{2} a^{2} - \frac{59}{2} a - 18\) , \( -\frac{2781}{2} a^{2} + \frac{10821}{2} a + 3834\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-6\right){x}^{2}+\left(\frac{15}{2}a^{2}-\frac{59}{2}a-18\right){x}-\frac{2781}{2}a^{2}+\frac{10821}{2}a+3834$ |
| 8.1-a2 |
8.1-a |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{49} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$16.04462735$ |
1.492523475 |
\( \frac{51416120837381}{2199023255552} a^{2} - \frac{720091938689771}{2199023255552} a + \frac{834832342065725}{549755813888} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 6\) , \( 0\) , \( -119759860 a^{2} + 465654682 a + 331717716\) , \( 1223694190430 a^{2} - 4758011815996 a - 3389457611904\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-6\right){x}^{2}+\left(-119759860a^{2}+465654682a+331717716\right){x}+1223694190430a^{2}-4758011815996a-3389457611904$ |
| 8.1-a3 |
8.1-a |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$2.005578419$ |
1.492523475 |
\( -\frac{49580783989077628301225}{64} a^{2} + \frac{80076517406253230096411}{64} a + \frac{40304886381043760786659}{4} \) |
\( \bigl[1\) , \( 0\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( -\frac{3075637441991}{2} a^{2} + \frac{11958804252095}{2} a + 4259537562131\) , \( -\frac{6157005899514058307}{2} a^{2} + \frac{23939892045151880463}{2} a + 8527012170282607165\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(-\frac{3075637441991}{2}a^{2}+\frac{11958804252095}{2}a+4259537562131\right){x}-\frac{6157005899514058307}{2}a^{2}+\frac{23939892045151880463}{2}a+8527012170282607165$ |
| 8.1-a4 |
8.1-a |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1$ |
$8.022313679$ |
1.492523475 |
\( -\frac{16539251112791}{2048} a^{2} + \frac{26320567871169}{2048} a + \frac{54220695855371}{512} \) |
\( \bigl[1\) , \( 0\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( -\frac{3095396232431}{2} a^{2} + \frac{12035631092575}{2} a + 4286902071651\) , \( -\frac{6075614882304537699}{2} a^{2} + \frac{23623424561241428639}{2} a + 8414291441145082861\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(-\frac{3095396232431}{2}a^{2}+\frac{12035631092575}{2}a+4286902071651\right){x}-\frac{6075614882304537699}{2}a^{2}+\frac{23623424561241428639}{2}a+8414291441145082861$ |
| 8.1-a5 |
8.1-a |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{17} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$2.005578419$ |
1.492523475 |
\( \frac{30420931852288651803}{256} a^{2} - \frac{59141881325312428975}{128} a - \frac{10532702922880478139}{32} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} - \frac{1}{2} a + 4\) , \( a\) , \( -158707903129450787 a^{2} + 617093783835962500 a + 439598156503851815\) , \( -99753682869079912702039732 a^{2} + 387865861746302664359408927 a + 276303411670367154257788225\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-\frac{1}{2}a^{2}-\frac{1}{2}a+4\right){x}^{2}+\left(-158707903129450787a^{2}+617093783835962500a+439598156503851815\right){x}-99753682869079912702039732a^{2}+387865861746302664359408927a+276303411670367154257788225$ |
| 8.1-a6 |
8.1-a |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{38} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$16.04462735$ |
1.492523475 |
\( -\frac{50621367819}{2097152} a^{2} + \frac{72238855205}{2097152} a + \frac{177831319381}{524288} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 5\) , \( a + 1\) , \( -15189325 a^{2} + 59059679 a + 42072259\) , \( -\frac{182505494641}{2} a^{2} + \frac{709624436071}{2} a + 252757037975\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+5\right){x}^{2}+\left(-15189325a^{2}+59059679a+42072259\right){x}-\frac{182505494641}{2}a^{2}+\frac{709624436071}{2}a+252757037975$ |
| 8.1-b1 |
8.1-b |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$2.005578419$ |
1.492523475 |
\( \frac{15247866708587800897593}{64} a^{2} + \frac{53417967852391853909671}{64} a + \frac{13543704268610741196775}{32} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( -\frac{3505619809}{2} a^{2} - \frac{8373456057}{2} a - 1912026652\) , \( -\frac{258349956520917}{2} a^{2} - \frac{1005318188491443}{2} a - 260302759228368\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){x}^{2}+\left(-\frac{3505619809}{2}a^{2}-\frac{8373456057}{2}a-1912026652\right){x}-\frac{258349956520917}{2}a^{2}-\frac{1005318188491443}{2}a-260302759228368$ |
| 8.1-b2 |
8.1-b |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1$ |
$8.022313679$ |
1.492523475 |
\( \frac{4890658379189}{2048} a^{2} + \frac{18406527088015}{2048} a + \frac{6182502630031}{1024} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( -\frac{219683109}{2} a^{2} - \frac{521078557}{2} a - 118695812\) , \( -\frac{4044776198393}{2} a^{2} - \frac{15678187411863}{2} a - 4056508189528\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){x}^{2}+\left(-\frac{219683109}{2}a^{2}-\frac{521078557}{2}a-118695812\right){x}-\frac{4044776198393}{2}a^{2}-\frac{15678187411863}{2}a-4056508189528$ |
| 8.1-b3 |
8.1-b |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{49} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$16.04462735$ |
1.492523475 |
\( -\frac{334337908926195}{2199023255552} a^{2} + \frac{900181485103823}{2199023255552} a + \frac{3264096924023135}{1099511627776} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( -758 a^{2} - 2697 a - 1366\) , \( -2163 a^{2} - 7356 a - 3714\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){x}^{2}+\left(-758a^{2}-2697a-1366\right){x}-2163a^{2}-7356a-3714$ |
| 8.1-b4 |
8.1-b |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{17} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$2.005578419$ |
1.492523475 |
\( -\frac{87862830798336206147}{512} a^{2} + \frac{141904764985854011229}{512} a + \frac{285699509166593758753}{128} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -136930219333 a^{2} - 480088861360 a - 243486540062\) , \( -120359968985897693 a^{2} - 421678979526764474 a - 213829021937955940\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-136930219333a^{2}-480088861360a-243486540062\right){x}-120359968985897693a^{2}-421678979526764474a-213829021937955940$ |
| 8.1-b5 |
8.1-b |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{38} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$16.04462735$ |
1.492523475 |
\( \frac{10808743693}{2097152} a^{2} + \frac{68816504559}{2097152} a + \frac{59320824895}{1048576} \) |
\( \bigl[1\) , \( a + 1\) , \( 0\) , \( -965526 a^{2} + 3725008 a + 2656418\) , \( -1189625710 a^{2} + 4621159606 a + 3292395148\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-965526a^{2}+3725008a+2656418\right){x}-1189625710a^{2}+4621159606a+3292395148$ |
| 8.1-b6 |
8.1-b |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{43} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$4.011156839$ |
1.492523475 |
\( \frac{4084086670437495}{4294967296} a^{2} - \frac{7939989176541817}{2147483648} a - \frac{1414047902813091}{536870912} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 5\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( -93008767623 a^{2} + 361640039403 a + 257620962662\) , \( -44749942048306662 a^{2} + 173998336065881803 a + 123950928971976024\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+5\right){x}^{2}+\left(-93008767623a^{2}+361640039403a+257620962662\right){x}-44749942048306662a^{2}+173998336065881803a+123950928971976024$ |
| 8.1-c1 |
8.1-c |
$2$ |
$3$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{21} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$158.3231545$ |
1.227311275 |
\( -\frac{8546212653}{262144} a^{2} - \frac{27793426991}{262144} a - \frac{6930928319}{131072} \) |
\( \bigl[1\) , \( 0\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( \frac{219}{2} a^{2} - \frac{355}{2} a - 1425\) , \( -140149 a^{2} + 226349 a + 1822858\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(\frac{219}{2}a^{2}-\frac{355}{2}a-1425\right){x}-140149a^{2}+226349a+1822858$ |
| 8.1-c2 |
8.1-c |
$2$ |
$3$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{55} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$5.863820537$ |
1.227311275 |
\( \frac{461685341165402387}{4503599627370496} a^{2} - \frac{1702134648083946351}{4503599627370496} a - \frac{610346759637856255}{2251799813685248} \) |
\( \bigl[1\) , \( 0\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( -988 a^{2} + 1595 a + 12850\) , \( 3783807 a^{2} - 6111121 a - 49214528\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(-988a^{2}+1595a+12850\right){x}+3783807a^{2}-6111121a-49214528$ |
| 8.1-d1 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1.547399842$ |
$6.007416618$ |
1.297098911 |
\( \frac{29264590191614421}{33554432} a^{2} + \frac{102522835692825317}{33554432} a + \frac{12996927008167565}{8388608} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 5\) , \( -7206 a^{2} - 25244 a - 12798\) , \( -1458099 a^{2} - 5108177 a - 2590294\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-5\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){x}^{2}+\left(-7206a^{2}-25244a-12798\right){x}-1458099a^{2}-5108177a-2590294$ |
| 8.1-d2 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.515799947$ |
$162.2002486$ |
1.297098911 |
\( \frac{9523381685}{512} a^{2} - \frac{37028757563}{512} a - \frac{6594534963}{128} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 5\) , \( -96 a^{2} - 334 a - 163\) , \( -\frac{3605}{2} a^{2} - \frac{12635}{2} a - 3210\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-5\right){y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){x}^{2}+\left(-96a^{2}-334a-163\right){x}-\frac{3605}{2}a^{2}-\frac{12635}{2}a-3210$ |
| 8.1-d3 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{36} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$3.094799685$ |
$12.01483323$ |
1.297098911 |
\( \frac{747081097}{4096} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( \frac{94881}{2} a^{2} - \frac{153241}{2} a - 617040\) , \( \frac{27023339}{2} a^{2} - \frac{43644637}{2} a - 175741102\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(\frac{94881}{2}a^{2}-\frac{153241}{2}a-617040\right){x}+\frac{27023339}{2}a^{2}-\frac{43644637}{2}a-175741102$ |
| 8.1-d4 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{12} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$1.031599895$ |
$324.4004973$ |
1.297098911 |
\( \frac{294937}{16} \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( \frac{6961}{2} a^{2} - \frac{11241}{2} a - 45265\) , \( -\frac{511731}{2} a^{2} + \frac{826483}{2} a + 3327943\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(\frac{6961}{2}a^{2}-\frac{11241}{2}a-45265\right){x}-\frac{511731}{2}a^{2}+\frac{826483}{2}a+3327943$ |
| 8.1-d5 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1.547399842$ |
$6.007416618$ |
1.297098911 |
\( -\frac{47579151566917145}{16777216} a^{2} + \frac{38421870879265783}{8388608} a + \frac{77355466306495533}{2097152} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 5\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 4\) , \( \frac{46861}{2} a^{2} - \frac{75685}{2} a - 304752\) , \( 4741237 a^{2} - 7657438 a - 61667460\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-4\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+5\right){x}^{2}+\left(\frac{46861}{2}a^{2}-\frac{75685}{2}a-304752\right){x}+4741237a^{2}-7657438a-61667460$ |
| 8.1-d6 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.515799947$ |
$162.2002486$ |
1.297098911 |
\( \frac{2114653127}{256} a^{2} + \frac{3704364279}{128} a + \frac{469740693}{32} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 5\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 4\) , \( \frac{621}{2} a^{2} - \frac{1005}{2} a - 4037\) , \( \frac{11725}{2} a^{2} - \frac{18941}{2} a - 76259\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-4\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+5\right){x}^{2}+\left(\frac{621}{2}a^{2}-\frac{1005}{2}a-4037\right){x}+\frac{11725}{2}a^{2}-\frac{18941}{2}a-76259$ |
| 8.1-d7 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{36} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \) |
$1.547399842$ |
$6.007416618$ |
1.297098911 |
\( \frac{65893712942219869}{33554432} a^{2} - \frac{256210319209888449}{33554432} a - \frac{91258046794472241}{16777216} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 6\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( -458920937794 a^{2} - 1607741225073 a - 815260204308\) , \( -5124405501726437361 a^{2} - 17952368917266193425 a - 9103362981097420174\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+6\right){x}^{2}+\left(-458920937794a^{2}-1607741225073a-815260204308\right){x}-5124405501726437361a^{2}-17952368917266193425a-9103362981097420174$ |
| 8.1-d8 |
8.1-d |
$8$ |
$12$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{12} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \) |
$0.515799947$ |
$162.2002486$ |
1.297098911 |
\( -\frac{13752687939}{512} a^{2} + \frac{22211300447}{512} a + \frac{89438590255}{256} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 6\) , \( \frac{1}{2} a^{2} + \frac{1}{2} a - 5\) , \( 50824917136 a^{2} + 178055320237 a + 90289043107\) , \( \frac{372704835802970103}{2} a^{2} + \frac{1305699696741374595}{2} a + 331049855830269162\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}+\frac{1}{2}a-5\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+6\right){x}^{2}+\left(50824917136a^{2}+178055320237a+90289043107\right){x}+\frac{372704835802970103}{2}a^{2}+\frac{1305699696741374595}{2}a+331049855830269162$ |
| 8.1-e1 |
8.1-e |
$2$ |
$3$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{55} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$5.863820537$ |
1.227311275 |
\( \frac{158539312293869595}{4503599627370496} a^{2} + \frac{764831370036935179}{4503599627370496} a + \frac{221849021777202659}{1125899906842624} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 684 a^{2} - 2660 a - 1895\) , \( -2620151 a^{2} + 10187764 a + 7257442\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(684a^{2}-2660a-1895\right){x}-2620151a^{2}+10187764a+7257442$ |
| 8.1-e2 |
8.1-e |
$2$ |
$3$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{21} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$158.3231545$ |
1.227311275 |
\( \frac{26716032475}{262144} a^{2} - \frac{43808457781}{262144} a - \frac{87347970653}{65536} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -76 a^{2} + 295 a + 210\) , \( 97048 a^{2} - 377347 a - 268810\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-76a^{2}+295a+210\right){x}+97048a^{2}-377347a-268810$ |
| 8.1-f1 |
8.1-f |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{49} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$16.04462735$ |
1.492523475 |
\( \frac{141460894044407}{1099511627776} a^{2} - \frac{45022386603513}{549755813888} a + \frac{57590028379613}{137438953472} \) |
\( \bigl[1\) , \( \frac{1}{2} a^{2} - \frac{3}{2} a - 6\) , \( 1\) , \( -231508319 a^{2} - 811041940 a - 411266247\) , \( \frac{9870738940249}{2} a^{2} + \frac{34580235098373}{2} a + 8767545763482\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{2}-\frac{3}{2}a-6\right){x}^{2}+\left(-231508319a^{2}-811041940a-411266247\right){x}+\frac{9870738940249}{2}a^{2}+\frac{34580235098373}{2}a+8767545763482$ |
| 8.1-f2 |
8.1-f |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{22} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1$ |
$8.022313679$ |
1.492523475 |
\( \frac{5824296366801}{1024} a^{2} - \frac{2795443434949}{128} a - \frac{997411025527}{64} \) |
\( \bigl[1\) , \( a + 1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( -17111084651837417087191121 a^{2} - 59945393498009613940367226 a - 30397363076259643751733794\) , \( -\frac{335141306007626193836066283971061515535}{2} a^{2} - \frac{1174103096024756590527747873015627175415}{2} a - 297684576046782321649388818426333171426\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-17111084651837417087191121a^{2}-59945393498009613940367226a-30397363076259643751733794\right){x}-\frac{335141306007626193836066283971061515535}{2}a^{2}-\frac{1174103096024756590527747873015627175415}{2}a-297684576046782321649388818426333171426$ |
| 8.1-f3 |
8.1-f |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$2.005578419$ |
1.492523475 |
\( \frac{2145807330030614212727}{4} a^{2} - \frac{66747242629322542003041}{32} a - \frac{11887157827602390982447}{8} \) |
\( \bigl[1\) , \( a + 1\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( -17001859754457708705606311 a^{2} - 59562745081124875228220796 a - 30203328102429877028223094\) , \( -\frac{339630973974282121554114567671113068355}{2} a^{2} - \frac{1189831784089408186615998173119700270387}{2} a - 301672460802516897349245344748210289374\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-17001859754457708705606311a^{2}-59562745081124875228220796a-30203328102429877028223094\right){x}-\frac{339630973974282121554114567671113068355}{2}a^{2}-\frac{1189831784089408186615998173119700270387}{2}a-301672460802516897349245344748210289374$ |
| 8.1-f4 |
8.1-f |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{38} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$16.04462735$ |
1.492523475 |
\( \frac{19906312063}{1048576} a^{2} - \frac{35263839941}{524288} a - \frac{5735744143}{131072} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{1}{2} a + 6\) , \( a + 1\) , \( -321602836036 a^{2} - 1126673670807 a - 571318439013\) , \( -\frac{843872945875866439}{2} a^{2} - \frac{2956346533965708663}{2} a - 749558337415711762\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{1}{2}a+6\right){x}^{2}+\left(-321602836036a^{2}-1126673670807a-571318439013\right){x}-\frac{843872945875866439}{2}a^{2}-\frac{2956346533965708663}{2}a-749558337415711762$ |
| 8.1-f5 |
8.1-f |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{17} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{3} \) |
$1$ |
$2.005578419$ |
1.492523475 |
\( \frac{27020967093758902541}{512} a^{2} + \frac{94662760315395704671}{512} a + \frac{24000995966470876607}{256} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 6\) , \( 1\) , \( -916219905552330547041395498513835829927 a^{2} - 3209800190144341109378871405995581840451 a - 1627639024261378952100318608657778286395\) , \( -\frac{131314141250980490650969942391720047214008258054469940702477}{2} a^{2} - \frac{460033833582721941020023768362835747799292951665803490641033}{2} a - 116637948729470886919472979143467987757090960807189124762352\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+6\right){x}^{2}+\left(-916219905552330547041395498513835829927a^{2}-3209800190144341109378871405995581840451a-1627639024261378952100318608657778286395\right){x}-\frac{131314141250980490650969942391720047214008258054469940702477}{2}a^{2}-\frac{460033833582721941020023768362835747799292951665803490641033}{2}a-116637948729470886919472979143467987757090960807189124762352$ |
| 8.1-f6 |
8.1-f |
$6$ |
$8$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{43} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$4.011156839$ |
1.492523475 |
\( -\frac{11795891682646139}{8589934592} a^{2} + \frac{19051328366188437}{8589934592} a + \frac{38356025106227389}{2147483648} \) |
\( \bigl[1\) , \( -\frac{1}{2} a^{2} + \frac{3}{2} a + 6\) , \( 1\) , \( \frac{5}{2} a^{2} + \frac{29}{2} a + 16\) , \( -\frac{1233}{2} a^{2} - \frac{4297}{2} a - 1084\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{2}+\frac{3}{2}a+6\right){x}^{2}+\left(\frac{5}{2}a^{2}+\frac{29}{2}a+16\right){x}-\frac{1233}{2}a^{2}-\frac{4297}{2}a-1084$ |
| 8.1-g1 |
8.1-g |
$2$ |
$3$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{21} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$158.3231545$ |
1.227311275 |
\( -\frac{9084909911}{131072} a^{2} + \frac{17900471193}{65536} a + \frac{2877160963}{16384} \) |
\( \bigl[1\) , \( 0\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( -34 a^{2} - 118 a - 58\) , \( 43101 a^{2} + 150995 a + 76564\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(-34a^{2}-118a-58\right){x}+43101a^{2}+150995a+76564$ |
| 8.1-g2 |
8.1-g |
$2$ |
$3$ |
3.3.1849.1 |
$3$ |
$[3, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{55} \) |
$5.43403$ |
$(3/2a^2-5/2a-19), (1/2a^2-1/2a-8), (-a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$5.863820537$ |
1.227311275 |
\( -\frac{310112326729635991}{2251799813685248} a^{2} + \frac{234325819511752793}{1125899906842624} a + \frac{522372320003280451}{281474976710656} \) |
\( \bigl[1\) , \( 0\) , \( \frac{1}{2} a^{2} - \frac{1}{2} a - 4\) , \( \frac{607}{2} a^{2} + \frac{2129}{2} a + 542\) , \( -1163656 a^{2} - 4076646 a - 2067206\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{2}a^{2}-\frac{1}{2}a-4\right){y}={x}^{3}+\left(\frac{607}{2}a^{2}+\frac{2129}{2}a+542\right){x}-1163656a^{2}-4076646a-2067206$ |
*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.
