| 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 |
$2$ |
$2$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{4} \) |
$1.12963$ |
$(-a+6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.42172449$ |
1.195737336 |
\( -\frac{4825}{16} a + \frac{28021}{16} \) |
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -57 a - 264\) , \( -14594 a - 70270\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-57a-264\right){x}-14594a-70270$ |
| 2.1-a2 |
2.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{2} \) |
$1.12963$ |
$(-a+6)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.42172449$ |
1.195737336 |
\( \frac{32125}{4} a + \frac{158187}{4} \) |
\( \bigl[1\) , \( -a - 1\) , \( a\) , \( 8\) , \( -a - 8\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+8{x}-a-8$ |
| 2.2-a1 |
2.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{2} \) |
$1.12963$ |
$(a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.42172449$ |
1.195737336 |
\( -\frac{32125}{4} a + 47578 \) |
\( \bigl[1\) , \( a + 1\) , \( a\) , \( a + 8\) , \( a - 1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+8\right){x}+a-1$ |
| 2.2-a2 |
2.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{4} \) |
$1.12963$ |
$(a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.42172449$ |
1.195737336 |
\( \frac{4825}{16} a + \frac{5799}{4} \) |
\( \bigl[1\) , \( a - 1\) , \( a\) , \( 56 a - 320\) , \( 14593 a - 84863\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(56a-320\right){x}+14593a-84863$ |
| 4.1-a1 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.625109551$ |
$39.87019907$ |
2.031751412 |
\( \frac{26287}{64} a + \frac{282725}{64} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 76 a - 435\) , \( -300 a + 1741\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(76a-435\right){x}-300a+1741$ |
| 4.1-a2 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.250219102$ |
$19.93509953$ |
2.031751412 |
\( \frac{49761}{64} a + \frac{79121}{16} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( -186564 a - 898307\) , \( -91146540 a - 438877227\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-186564a-898307\right){x}-91146540a-438877227$ |
| 4.1-a3 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{19} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.875328653$ |
$4.430022119$ |
2.031751412 |
\( -\frac{223965069241}{262144} a + \frac{1515273464805}{262144} \) |
\( \bigl[1\) , \( -a + 1\) , \( 0\) , \( 2861 a - 16630\) , \( 197859 a - 1150568\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2861a-16630\right){x}+197859a-1150568$ |
| 4.1-a4 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{11} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$9.750657307$ |
$2.215011059$ |
2.031751412 |
\( \frac{31710781726929}{512} a + \frac{152689724672611}{512} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 2260547 a - 13145249\) , \( 5564451284 a - 32357689912\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(2260547a-13145249\right){x}+5564451284a-32357689912$ |
| 4.1-b1 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{3} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$49.02101037$ |
1.024779721 |
\( -\frac{11575}{4} a + \frac{78637}{4} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 3166 a - 18410\) , \( -216410 a + 1258440\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(3166a-18410\right){x}-216410a+1258440$ |
| 4.1-b2 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{3} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$4$ |
\( 2 \) |
$1$ |
$49.02101037$ |
1.024779721 |
\( \frac{11575}{4} a + \frac{33531}{2} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -3166 a - 15244\) , \( 216410 a + 1042030\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-3166a-15244\right){x}+216410a+1042030$ |
| 4.1-b3 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \) |
$1$ |
$5.446778930$ |
1.024779721 |
\( -\frac{46947989887}{64} a + \frac{68252920837}{16} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -34016 a - 163789\) , \( -7810456 a - 37607915\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-34016a-163789\right){x}-7810456a-37607915$ |
| 4.1-b4 |
4.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$4$ |
\( 2 \) |
$1$ |
$5.446778930$ |
1.024779721 |
\( \frac{46947989887}{64} a + \frac{226063693461}{64} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 34016 a - 197805\) , \( 7810456 a - 45418371\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(34016a-197805\right){x}+7810456a-45418371$ |
| 4.1-c1 |
4.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$17.82077109$ |
3.352874252 |
\( -\frac{1178793}{256} a + \frac{6622101}{256} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 2 a - 12\) , \( -4 a + 23\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(2a-12\right){x}-4a+23$ |
| 4.1-c2 |
4.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$4.455192774$ |
3.352874252 |
\( -\frac{2080148815899}{4} a + \frac{6048108512703}{2} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 12122 a + 58368\) , \( -1091464 a - 5255479\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(12122a+58368\right){x}-1091464a-5255479$ |
| 4.1-c3 |
4.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$17.82077109$ |
3.352874252 |
\( -\frac{3536379}{16} a + 1287684 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -3668 a - 17662\) , \( -147784 a - 711591\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-3668a-17662\right){x}-147784a-711591$ |
| 4.1-c4 |
4.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$17.82077109$ |
3.352874252 |
\( \frac{1178793}{256} a + \frac{1360827}{64} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -2 a - 10\) , \( 4 a + 19\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-2a-10\right){x}+4a+19$ |
| 4.1-c5 |
4.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{6} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$17.82077109$ |
3.352874252 |
\( \frac{3536379}{16} a + \frac{17066565}{16} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 3668 a - 21330\) , \( 147784 a - 859375\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(3668a-21330\right){x}+147784a-859375$ |
| 4.1-c6 |
4.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{3} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$4.455192774$ |
3.352874252 |
\( \frac{2080148815899}{4} a + \frac{10016068209507}{4} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -12122 a + 70490\) , \( 1091464 a - 6346943\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-12122a+70490\right){x}+1091464a-6346943$ |
| 4.1-d1 |
4.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{12} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$34.56484808$ |
0.361287487 |
\( -\frac{85779161}{256} a + \frac{498494325}{256} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 27992 a - 162776\) , \( -5873416 a + 34154342\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(27992a-162776\right){x}-5873416a+34154342$ |
| 4.1-d2 |
4.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{36} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.840538676$ |
0.361287487 |
\( -\frac{4065896329}{16777216} a + \frac{37338003829}{16777216} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 74632 a - 433991\) , \( 17928072 a - 104253046\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(74632a-433991\right){x}+17928072a-104253046$ |
| 4.1-d3 |
4.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{36} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.840538676$ |
0.361287487 |
\( \frac{4065896329}{16777216} a + \frac{8318026875}{4194304} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -74632 a - 359359\) , \( -17928072 a - 86324974\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-74632a-359359\right){x}-17928072a-86324974$ |
| 4.1-d4 |
4.1-d |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{12} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$34.56484808$ |
0.361287487 |
\( \frac{85779161}{256} a + \frac{103178791}{64} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -27992 a - 134784\) , \( 5873416 a + 28280926\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-27992a-134784\right){x}+5873416a+28280926$ |
| 4.1-e1 |
4.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{11} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$9.750657307$ |
$2.215011059$ |
2.031751412 |
\( -\frac{31710781726929}{512} a + \frac{46100126599885}{128} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -2260548 a - 10884701\) , \( -5564451285 a - 26793238627\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-2260548a-10884701\right){x}-5564451285a-26793238627$ |
| 4.1-e2 |
4.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$3.250219102$ |
$19.93509953$ |
2.031751412 |
\( -\frac{49761}{64} a + \frac{366245}{64} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 186564 a - 1084871\) , \( 91146540 a - 530023767\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(186564a-1084871\right){x}+91146540a-530023767$ |
| 4.1-e3 |
4.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{9} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1.625109551$ |
$39.87019907$ |
2.031751412 |
\( -\frac{26287}{64} a + \frac{77253}{16} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -76 a - 359\) , \( 300 a + 1441\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-76a-359\right){x}+300a+1441$ |
| 4.1-e4 |
4.1-e |
$4$ |
$6$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{19} \) |
$1.34336$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$4.875328653$ |
$4.430022119$ |
2.031751412 |
\( \frac{223965069241}{262144} a + \frac{322827098891}{65536} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( -2861 a - 13769\) , \( -197859 a - 952709\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(-2861a-13769\right){x}-197859a-952709$ |
| 7.1-a1 |
7.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
7.1 |
\( 7 \) |
\( -7 \) |
$1.54508$ |
$(-6a+35)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.105609247$ |
$57.33100042$ |
1.139153486 |
\( \frac{56309}{7} a - \frac{314609}{7} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 542 a - 3042\) , \( -15560 a + 90714\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(542a-3042\right){x}-15560a+90714$ |
| 7.2-a1 |
7.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
7.2 |
\( 7 \) |
\( -7 \) |
$1.54508$ |
$(6a+29)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.105609247$ |
$57.33100042$ |
1.139153486 |
\( -\frac{56309}{7} a - 36900 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -526 a - 2530\) , \( 13031 a + 62745\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-526a-2530\right){x}+13031a+62745$ |
| 8.3-a1 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.59753$ |
$(a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1.346261804$ |
$23.23685671$ |
1.471423497 |
\( -171 a + 2669 \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -a + 18\) , \( -a + 15\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-a+18\right){x}-a+15$ |
| 8.3-a2 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.59753$ |
$(a+5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$2.692523609$ |
$23.23685671$ |
1.471423497 |
\( -1989 a + 13349 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 485 a - 2806\) , \( 13245 a - 76988\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(485a-2806\right){x}+13245a-76988$ |
| 8.3-a3 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( 2^{11} \) |
$1.59753$ |
$(a+5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$5.385047219$ |
$5.809214178$ |
1.471423497 |
\( -46235647 a + 269026053 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 7650 a - 44471\) , \( 846103 a - 4920118\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(7650a-44471\right){x}+846103a-4920118$ |
| 8.3-a4 |
8.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{8} \) |
$1.59753$ |
$(a+5)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.346261804$ |
$23.23685671$ |
1.471423497 |
\( 49249 a + 237155 \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -140228 a - 675213\) , \( 65294067 a + 314395690\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-140228a-675213\right){x}+65294067a+314395690$ |
| 8.4-a1 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{8} \) |
$1.59753$ |
$(-a+6)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.346261804$ |
$23.23685671$ |
1.471423497 |
\( -49249 a + 286404 \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 140226 a - 815439\) , \( -65294068 a + 379689758\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(140226a-815439\right){x}-65294068a+379689758$ |
| 8.4-a2 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.59753$ |
$(-a+6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1.346261804$ |
$23.23685671$ |
1.471423497 |
\( 171 a + 2498 \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( a + 17\) , \( a + 14\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+17\right){x}+a+14$ |
| 8.4-a3 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{10} \) |
$1.59753$ |
$(-a+6)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$2.692523609$ |
$23.23685671$ |
1.471423497 |
\( 1989 a + 11360 \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -487 a - 2319\) , \( -13246 a - 63742\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-487a-2319\right){x}-13246a-63742$ |
| 8.4-a4 |
8.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( 2^{11} \) |
$1.59753$ |
$(-a+6)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$5.385047219$ |
$5.809214178$ |
1.471423497 |
\( 46235647 a + 222790406 \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -7652 a - 36819\) , \( -846104 a - 4074014\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-7652a-36819\right){x}-846104a-4074014$ |
| 9.1-a1 |
9.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$1.64527$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.553806322$ |
$13.51799551$ |
3.951845490 |
\( -\frac{4096}{3} a - \frac{16384}{3} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9 a - 43\) , \( -39 a - 188\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(-9a-43\right){x}-39a-188$ |
| 9.1-b1 |
9.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$1.64527$ |
$(3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1.553806322$ |
$13.51799551$ |
3.951845490 |
\( \frac{4096}{3} a - \frac{20480}{3} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 9 a - 52\) , \( 39 a - 227\bigr] \) |
${y}^2+{y}={x}^{3}+{x}^{2}+\left(9a-52\right){x}+39a-227$ |
| 14.1-a1 |
14.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{24} \cdot 7^{3} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$4.172683519$ |
3.140264371 |
\( \frac{1148037253661}{5754585088} a + \frac{5489063415559}{5754585088} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -1886 a - 9086\) , \( -51323 a - 247128\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-1886a-9086\right){x}-51323a-247128$ |
| 14.1-a2 |
14.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{8} \cdot 7^{9} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$1$ |
$0.463631502$ |
3.140264371 |
\( \frac{1665135701693794453}{10330523392} a + \frac{7663336478308515567}{10330523392} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -127126 a - 612126\) , \( -57375186 a - 276265708\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-127126a-612126\right){x}-57375186a-276265708$ |
| 14.1-b1 |
14.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{2} \cdot 7^{3} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$4.796383962$ |
0.902411697 |
\( \frac{182510625893}{1372} a - \frac{1061308646049}{1372} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 69 a - 308\) , \( 592 a - 3220\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(69a-308\right){x}+592a-3220$ |
| 14.1-c1 |
14.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7^{4} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$10.33041057$ |
0.971803280 |
\( -\frac{41477305}{153664} a + \frac{165712549}{153664} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( 67 a - 393\) , \( -359 a + 2076\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(67a-393\right){x}-359a+2076$ |
| 14.1-c2 |
14.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{3} \cdot 7^{2} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$20.66082114$ |
0.971803280 |
\( -\frac{39961897}{392} a + \frac{680780061}{392} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( 794947 a - 4622663\) , \( -890320822 a + 5177280498\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(794947a-4622663\right){x}-890320822a+5177280498$ |
| 14.1-d1 |
14.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2 \cdot 7 \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.568513406$ |
$26.90542464$ |
2.877871081 |
\( \frac{4591}{14} a - \frac{26697}{14} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( -a - 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-a-5$ |
| 14.1-e1 |
14.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{6} \cdot 7 \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.466024851$ |
$13.11529653$ |
6.899684234 |
\( \frac{307253}{448} a + \frac{1346511}{448} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -17096 a - 82315\) , \( -2529384 a - 12179169\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-17096a-82315\right){x}-2529384a-12179169$ |
| 14.1-f1 |
14.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{36} \cdot 7^{2} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$0.959190705$ |
$2.886487882$ |
2.344110003 |
\( -\frac{37781419325932825}{3367254360064} a + \frac{219737109836120517}{3367254360064} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 1898040 a + 9139201\) , \( 41942564345 a + 201956505201\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}+\left(1898040a+9139201\right){x}+41942564345a+201956505201$ |
| 14.1-f2 |
14.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{9} \cdot 7^{2} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$3.836762821$ |
$0.721621970$ |
2.344110003 |
\( -\frac{465964359967014611784553}{25088} a + \frac{2709616724976978024576469}{25088} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -13559 a - 65281\) , \( -1332816 a - 6417625\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-13559a-65281\right){x}-1332816a-6417625$ |
| 14.1-f3 |
14.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( 2^{18} \cdot 7^{4} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1.918381410$ |
$2.886487882$ |
2.344110003 |
\( -\frac{831964927268316585}{629407744} a + \frac{4837953108983406869}{629407744} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -5274 a - 25381\) , \( 472335 a + 2274331\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-5274a-25381\right){x}+472335a+2274331$ |
| 14.1-f4 |
14.1-f |
$4$ |
$4$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.1 |
\( 2 \cdot 7 \) |
\( - 2^{9} \cdot 7^{8} \) |
$1.83743$ |
$(-a+6), (-6a+35)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3^{2} \) |
$0.959190705$ |
$2.886487882$ |
2.344110003 |
\( \frac{177853984311731577}{2951578112} a + \frac{856032218965371739}{2951578112} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 32 a - 251\) , \( 319 a - 1659\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(32a-251\right){x}+319a-1659$ |
| 14.2-a1 |
14.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{113}) \) |
$2$ |
$[2, 0]$ |
14.2 |
\( 2 \cdot 7 \) |
\( 2^{9} \cdot 7 \) |
$1.83743$ |
$(-a+6), (6a+29)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3^{2} \) |
$1$ |
$7.555541069$ |
6.396889640 |
\( \frac{33979}{3584} a - \frac{28233}{512} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( 1\) , \( 4 a - 19\) , \( 82 a - 473\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a-19\right){x}+82a-473$ |
*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.
