| 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 |
| 15.2-a1 |
15.2-a |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5 \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 5 \) |
$0.310847401$ |
$67.18745005$ |
3.265980363 |
\( \frac{20022119795364993979}{1215} a^{3} + \frac{50015188527258048512}{1215} a^{2} + \frac{4805975411558531482}{1215} a - \frac{8014983840650153123}{1215} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( -a^{3} - a^{2} + 6 a + 3\) , \( a^{3} - 6 a - 1\) , \( -108 a^{3} + 22 a^{2} + 623 a - 19\) , \( -357 a^{3} + 296 a^{2} + 2035 a - 1331\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(-a^{3}-a^{2}+6a+3\right){x}^{2}+\left(-108a^{3}+22a^{2}+623a-19\right){x}-357a^{3}+296a^{2}+2035a-1331$ |
| 15.2-a2 |
15.2-a |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{10} \cdot 5^{2} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.621694803$ |
$33.59372502$ |
3.265980363 |
\( -\frac{2759330739183809441224321}{1476225} a^{3} + \frac{6402088664839967667369712}{1476225} a^{2} + \frac{1701964657768270062491237}{1476225} a - \frac{1189201702748248621685893}{1476225} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( 1\) , \( 340 a^{3} - 681 a^{2} - 415 a\) , \( -9454 a^{3} + 22762 a^{2} + 4270 a - 5088\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(340a^{3}-681a^{2}-415a\right){x}-9454a^{3}+22762a^{2}+4270a-5088$ |
| 15.2-a3 |
15.2-a |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{20} \cdot 5^{4} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.310847401$ |
$134.3749001$ |
3.265980363 |
\( -\frac{623401384142926281989}{2179240250625} a^{3} + \frac{1436465931996585003008}{2179240250625} a^{2} + \frac{404228997013698367258}{2179240250625} a - \frac{255920592800296711187}{2179240250625} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( 1\) , \( 20 a^{3} - 46 a^{2} - 25 a + 5\) , \( -148 a^{3} + 376 a^{2} + 58 a - 90\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(20a^{3}-46a^{2}-25a+5\right){x}-148a^{3}+376a^{2}+58a-90$ |
| 15.2-a4 |
15.2-a |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{10} \cdot 5^{2} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 5 \) |
$0.155423700$ |
$537.4996004$ |
3.265980363 |
\( \frac{34024284020071}{1476225} a^{3} + \frac{84422484313463}{1476225} a^{2} + \frac{6992362655638}{1476225} a - \frac{13155841683257}{1476225} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( 1\) , \( -6 a^{2} + 5\) , \( 14 a^{2} + 4 a - 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-6a^{2}+5\right){x}+14a^{2}+4a-2$ |
| 15.2-a5 |
15.2-a |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5 \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 5 \) |
$0.077711850$ |
$1074.999200$ |
3.265980363 |
\( -\frac{6257059}{1215} a^{3} - \frac{4227017}{1215} a^{2} + \frac{30638723}{1215} a + \frac{15689258}{1215} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( 1\) , \( -a^{2} + 5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-a^{2}+5\right){x}+2$ |
| 15.2-a6 |
15.2-a |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{40} \cdot 5^{8} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \cdot 5 \) |
$0.621694803$ |
$8.398431257$ |
3.265980363 |
\( \frac{20823095928489856316807293441}{4749088069944112812890625} a^{3} - \frac{37244638197749598212307123952}{4749088069944112812890625} a^{2} - \frac{32444176320962389804536879077}{4749088069944112812890625} a + \frac{3780218757573537123370935253}{4749088069944112812890625} \) |
\( \bigl[1\) , \( -a^{2} + 4\) , \( 1\) , \( 20 a^{3} - 51 a^{2} - 35 a + 10\) , \( -154 a^{3} + 358 a^{2} + 42 a - 84\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(20a^{3}-51a^{2}-35a+10\right){x}-154a^{3}+358a^{2}+42a-84$ |
| 15.2-b1 |
15.2-b |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.092267086$ |
$459.4075221$ |
3.977171596 |
\( -\frac{141544361317964}{18225} a^{3} + \frac{328404077969408}{18225} a^{2} + \frac{87307968145483}{18225} a - \frac{61000042547837}{18225} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 3\) , \( -a^{2} - a + 3\) , \( a^{3} + a^{2} - 5 a - 3\) , \( 4 a^{3} - 2 a^{2} - 26 a - 9\) , \( 22 a^{3} + 6 a^{2} - 137 a - 68\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-3\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(4a^{3}-2a^{2}-26a-9\right){x}+22a^{3}+6a^{2}-137a-68$ |
| 15.2-b2 |
15.2-b |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5 \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.184534173$ |
$918.8150443$ |
3.977171596 |
\( \frac{2437304}{135} a^{3} - \frac{5743313}{135} a^{2} - \frac{1304218}{135} a + \frac{1230887}{135} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 3\) , \( -a^{2} - a + 3\) , \( a^{3} + a^{2} - 5 a - 3\) , \( -a^{3} - 2 a^{2} + 4 a + 6\) , \( -a^{3} - a^{2} + 5 a + 4\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-3\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-a^{3}-2a^{2}+4a+6\right){x}-a^{3}-a^{2}+5a+4$ |
| 15.2-b3 |
15.2-b |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{18} \cdot 5^{6} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.030755695$ |
$459.4075221$ |
3.977171596 |
\( \frac{2415272784052678096}{6053445140625} a^{3} + \frac{1575589929141641138}{6053445140625} a^{2} - \frac{8258181364559086487}{6053445140625} a + \frac{2563578429804827368}{6053445140625} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 5 a + 1\) , \( a^{3} + a^{2} - 5 a - 4\) , \( -14 a^{3} + 13 a^{2} + 77 a - 66\) , \( 74 a^{3} - 42 a^{2} - 425 a + 164\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{3}+a^{2}-5a-4\right){y}={x}^{3}+\left(-a^{3}+5a+1\right){x}^{2}+\left(-14a^{3}+13a^{2}+77a-66\right){x}+74a^{3}-42a^{2}-425a+164$ |
| 15.2-b4 |
15.2-b |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3^{2} \) |
$0.061511391$ |
$918.8150443$ |
3.977171596 |
\( -\frac{40896025306276}{2460375} a^{3} - \frac{13735726285553}{2460375} a^{2} + \frac{240761734608572}{2460375} a + \frac{121759468842692}{2460375} \) |
\( \bigl[a^{3} - 5 a\) , \( -a^{3} - a^{2} + 5 a + 5\) , \( 0\) , \( -3 a^{3} - a^{2} + 15 a + 12\) , \( -3 a^{3} + 2 a^{2} + 14 a + 8\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}={x}^{3}+\left(-a^{3}-a^{2}+5a+5\right){x}^{2}+\left(-3a^{3}-a^{2}+15a+12\right){x}-3a^{3}+2a^{2}+14a+8$ |
| 15.2-c1 |
15.2-c |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{10} \cdot 5^{2} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$9.232141111$ |
$2.477833980$ |
2.861823161 |
\( -\frac{2759330739183809441224321}{1476225} a^{3} + \frac{6402088664839967667369712}{1476225} a^{2} + \frac{1701964657768270062491237}{1476225} a - \frac{1189201702748248621685893}{1476225} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( 420 a^{3} + 5 a^{2} - 2180 a - 1129\) , \( 7468 a^{3} + 400 a^{2} - 39723 a - 19662\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(420a^{3}+5a^{2}-2180a-1129\right){x}+7468a^{3}+400a^{2}-39723a-19662$ |
| 15.2-c2 |
15.2-c |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{20} \cdot 5^{4} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.616070555$ |
$39.64534368$ |
2.861823161 |
\( -\frac{623401384142926281989}{2179240250625} a^{3} + \frac{1436465931996585003008}{2179240250625} a^{2} + \frac{404228997013698367258}{2179240250625} a - \frac{255920592800296711187}{2179240250625} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( 25 a^{3} - 130 a - 69\) , \( 140 a^{3} + 9 a^{2} - 748 a - 362\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(25a^{3}-130a-69\right){x}+140a^{3}+9a^{2}-748a-362$ |
| 15.2-c3 |
15.2-c |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{40} \cdot 5^{8} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$9.232141111$ |
$2.477833980$ |
2.861823161 |
\( \frac{20823095928489856316807293441}{4749088069944112812890625} a^{3} - \frac{37244638197749598212307123952}{4749088069944112812890625} a^{2} - \frac{32444176320962389804536879077}{4749088069944112812890625} a + \frac{3780218757573537123370935253}{4749088069944112812890625} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( 30 a^{3} - 5 a^{2} - 160 a - 49\) , \( 152 a^{3} + 14 a^{2} - 821 a - 410\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(30a^{3}-5a^{2}-160a-49\right){x}+152a^{3}+14a^{2}-821a-410$ |
| 15.2-c4 |
15.2-c |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5 \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$4.616070555$ |
$317.1627494$ |
2.861823161 |
\( \frac{20022119795364993979}{1215} a^{3} + \frac{50015188527258048512}{1215} a^{2} + \frac{4805975411558531482}{1215} a - \frac{8014983840650153123}{1215} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( -25 a^{3} + 130 a - 19\) , \( 90 a^{3} - 9 a^{2} - 448 a + 132\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+\left(-25a^{3}+130a-19\right){x}+90a^{3}-9a^{2}-448a+132$ |
| 15.2-c5 |
15.2-c |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{10} \cdot 5^{2} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.308035277$ |
$634.3254989$ |
2.861823161 |
\( \frac{34024284020071}{1476225} a^{3} + \frac{84422484313463}{1476225} a^{2} + \frac{6992362655638}{1476225} a - \frac{13155841683257}{1476225} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( -4\) , \( 5 a^{3} - 26 a - 5\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}-4{x}+5a^{3}-26a-5$ |
| 15.2-c6 |
15.2-c |
$6$ |
$8$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{5} \cdot 5 \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1.154017638$ |
$1268.650997$ |
2.861823161 |
\( -\frac{6257059}{1215} a^{3} - \frac{4227017}{1215} a^{2} + \frac{30638723}{1215} a + \frac{15689258}{1215} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 5 a\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}={x}^{3}+\left(-a^{3}+5a\right){x}^{2}+{x}$ |
| 15.2-d1 |
15.2-d |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 2^{2} \) |
$1$ |
$150.5685044$ |
2.886872071 |
\( -\frac{141544361317964}{18225} a^{3} + \frac{328404077969408}{18225} a^{2} + \frac{87307968145483}{18225} a - \frac{61000042547837}{18225} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{3} + 6 a\) , \( a^{3} + a^{2} - 6 a - 4\) , \( 2 a^{3} - 7 a^{2} - 8 a - 2\) , \( -3 a^{3} - 34 a^{2} - 20 a - 2\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-6a-4\right){y}={x}^{3}+\left(-a^{3}+6a\right){x}^{2}+\left(2a^{3}-7a^{2}-8a-2\right){x}-3a^{3}-34a^{2}-20a-2$ |
| 15.2-d2 |
15.2-d |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( 3^{18} \cdot 5^{6} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 2^{2} \cdot 3 \) |
$1$ |
$150.5685044$ |
2.886872071 |
\( \frac{2415272784052678096}{6053445140625} a^{3} + \frac{1575589929141641138}{6053445140625} a^{2} - \frac{8258181364559086487}{6053445140625} a + \frac{2563578429804827368}{6053445140625} \) |
\( \bigl[a\) , \( -a^{2} - a + 4\) , \( 0\) , \( -4 a^{3} + 24 a - 8\) , \( -16 a^{3} + 9 a^{2} + 105 a - 36\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(-4a^{3}+24a-8\right){x}-16a^{3}+9a^{2}+105a-36$ |
| 15.2-d3 |
15.2-d |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{9} \cdot 5^{3} \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 3 \) |
$1$ |
$301.1370088$ |
2.886872071 |
\( -\frac{40896025306276}{2460375} a^{3} - \frac{13735726285553}{2460375} a^{2} + \frac{240761734608572}{2460375} a + \frac{121759468842692}{2460375} \) |
\( \bigl[a\) , \( -a^{2} - a + 4\) , \( 0\) , \( a^{3} - 6 a + 2\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(a^{3}-6a+2\right){x}$ |
| 15.2-d4 |
15.2-d |
$4$ |
$6$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
15.2 |
\( 3 \cdot 5 \) |
\( - 3^{3} \cdot 5 \) |
$16.03252$ |
$(-a-1), (a^2+2a)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
|
\( 1 \) |
$1$ |
$301.1370088$ |
2.886872071 |
\( \frac{2437304}{135} a^{3} - \frac{5743313}{135} a^{2} - \frac{1304218}{135} a + \frac{1230887}{135} \) |
\( \bigl[a + 1\) , \( -a^{2} - a + 2\) , \( a^{3} - 5 a - 1\) , \( 4 a^{3} + a^{2} - 23 a - 10\) , \( 4 a^{3} + a^{2} - 24 a - 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(4a^{3}+a^{2}-23a-10\right){x}+4a^{3}+a^{2}-24a-12$ |
| 25.2-a1 |
25.2-a |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$172.5411210$ |
1.349089582 |
\( \frac{37687449}{125} a^{3} - \frac{19419931}{125} a^{2} - \frac{43245961}{25} a + \frac{73603996}{125} \) |
\( \bigl[a^{3} - 6 a - 1\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{3} - 6 a - 1\) , \( -123 a^{3} - 42 a^{2} + 724 a + 375\) , \( -109 a^{3} - 39 a^{2} + 644 a + 332\bigr] \) |
${y}^2+\left(a^{3}-6a-1\right){x}{y}+\left(a^{3}-6a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-123a^{3}-42a^{2}+724a+375\right){x}-109a^{3}-39a^{2}+644a+332$ |
| 25.2-a2 |
25.2-a |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$172.5411210$ |
1.349089582 |
\( \frac{50670520594}{1953125} a^{3} - \frac{100307515486}{1953125} a^{2} - \frac{13618694406}{390625} a + \frac{8341092601}{1953125} \) |
\( \bigl[a^{3} - 6 a\) , \( a^{3} - 7 a\) , \( a^{2} + a - 2\) , \( 5 a^{3} + 2 a^{2} - 24 a + 1\) , \( a^{2} + 4 a + 6\bigr] \) |
${y}^2+\left(a^{3}-6a\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(a^{3}-7a\right){x}^{2}+\left(5a^{3}+2a^{2}-24a+1\right){x}+a^{2}+4a+6$ |
| 25.2-b1 |
25.2-b |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.086899803$ |
$229.6464956$ |
2.496587376 |
\( -\frac{716724786161}{390625} a^{3} + \frac{1442006043167}{390625} a^{2} + \frac{880238515792}{390625} a - \frac{24734039638}{390625} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( -6 a^{2} + 14 a - 1\) , \( -15 a^{3} + 33 a^{2} + 13 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-6a^{2}+14a-1\right){x}-15a^{3}+33a^{2}+13a-7$ |
| 25.2-b2 |
25.2-b |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.043449901$ |
$918.5859824$ |
2.496587376 |
\( \frac{194297}{625} a^{3} - \frac{281609}{625} a^{2} + \frac{263541}{625} a + \frac{1172301}{625} \) |
\( \bigl[a + 1\) , \( -a^{2} + a + 3\) , \( a + 1\) , \( -a^{2} - a + 4\) , \( -2 a + 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+a+3\right){x}^{2}+\left(-a^{2}-a+4\right){x}-2a+2$ |
| 25.2-c1 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{16} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$19.76098557$ |
2.472161035 |
\( \frac{9389615014508350332}{390625} a^{3} - \frac{4823466677992809061}{390625} a^{2} - \frac{53859864331393680042}{390625} a + \frac{18278320539983543932}{390625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( -102 a^{3} - 21 a^{2} + 795 a - 268\) , \( -1965 a^{3} + 1912 a^{2} + 8713 a - 3073\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-102a^{3}-21a^{2}+795a-268\right){x}-1965a^{3}+1912a^{2}+8713a-3073$ |
| 25.2-c2 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$1264.703077$ |
2.472161035 |
\( -\frac{632164}{25} a^{3} - \frac{867543}{5} a^{2} + \frac{12706826}{25} a + \frac{7522477}{25} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( 3 a^{3} - 6 a^{2} - 5 a - 3\) , \( -9 a^{3} + 19 a^{2} + 7 a - 5\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(3a^{3}-6a^{2}-5a-3\right){x}-9a^{3}+19a^{2}+7a-5$ |
| 25.2-c3 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$316.1757692$ |
2.472161035 |
\( -\frac{12122311834764}{125} a^{3} + \frac{140632327637578}{625} a^{2} + \frac{37375658851457}{625} a - \frac{26121479320169}{625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( a^{3} - 5 a - 2\) , \( a^{3} + a^{2} - 5 a - 3\) , \( 33 a^{3} - 91 a^{2} + 20 a - 3\) , \( -524 a^{3} + 1171 a^{2} + 446 a - 263\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+\left(a^{3}+a^{2}-5a-3\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(33a^{3}-91a^{2}+20a-3\right){x}-524a^{3}+1171a^{2}+446a-263$ |
| 25.2-c4 |
25.2-c |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$79.04394231$ |
2.472161035 |
\( -\frac{5435814121359579324398236}{25} a^{3} + \frac{2522413903749638819346713}{5} a^{2} + \frac{3352607911018460603904474}{25} a - \frac{2342841126751495875736652}{25} \) |
\( \bigl[a^{2} + a - 2\) , \( -a^{2} + 3\) , \( a^{3} - 6 a\) , \( -128 a^{3} + 15 a^{2} + 856 a - 217\) , \( -1453 a^{3} + 1120 a^{2} + 7237 a - 2399\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-128a^{3}+15a^{2}+856a-217\right){x}-1453a^{3}+1120a^{2}+7237a-2399$ |
| 25.2-d1 |
25.2-d |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{2} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$0 \le r \le 1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
|
\( 1 \) |
$1$ |
$105.4074807$ |
6.378573759 |
\( \frac{1735607832}{5} a^{3} + \frac{582935254}{5} a^{2} - \frac{10217847707}{5} a - \frac{5167465863}{5} \) |
\( \bigl[a^{3} - 5 a\) , \( -a^{2} - a + 2\) , \( a^{2} - 2\) , \( -5 a^{3} + a^{2} + 33 a - 11\) , \( 4 a^{3} - 6 a^{2} - 15 a + 5\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}-a+2\right){x}^{2}+\left(-5a^{3}+a^{2}+33a-11\right){x}+4a^{3}-6a^{2}-15a+5$ |
| 25.2-e1 |
25.2-e |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{2} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$229.5262754$ |
1.794653385 |
\( \frac{1735607832}{5} a^{3} + \frac{582935254}{5} a^{2} - \frac{10217847707}{5} a - \frac{5167465863}{5} \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{2} + a - 2\) , \( -28 a^{3} + 14 a^{2} + 158 a - 54\) , \( -97 a^{3} + 49 a^{2} + 554 a - 189\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-28a^{3}+14a^{2}+158a-54\right){x}-97a^{3}+49a^{2}+554a-189$ |
| 25.2-f1 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{16} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.606791681$ |
$44.94626835$ |
3.411940230 |
\( \frac{9389615014508350332}{390625} a^{3} - \frac{4823466677992809061}{390625} a^{2} - \frac{53859864331393680042}{390625} a + \frac{18278320539983543932}{390625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( -536 a^{3} + 263 a^{2} + 3111 a - 1068\) , \( 9807 a^{3} - 5106 a^{2} - 56036 a + 18963\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-536a^{3}+263a^{2}+3111a-1068\right){x}+9807a^{3}-5106a^{2}-56036a+18963$ |
| 25.2-f2 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{8} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.213583362$ |
$179.7850734$ |
3.411940230 |
\( -\frac{12122311834764}{125} a^{3} + \frac{140632327637578}{625} a^{2} + \frac{37375658851457}{625} a - \frac{26121479320169}{625} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( -26 a^{3} - 2 a^{2} + 186 a - 58\) , \( 155 a^{3} - 159 a^{2} - 669 a + 239\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-26a^{3}-2a^{2}+186a-58\right){x}+155a^{3}-159a^{2}-669a+239$ |
| 25.2-f3 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$2.427166725$ |
$11.23656708$ |
3.411940230 |
\( -\frac{5435814121359579324398236}{25} a^{3} + \frac{2522413903749638819346713}{5} a^{2} + \frac{3352607911018460603904474}{25} a - \frac{2342841126751495875736652}{25} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( 84 a^{3} - 267 a^{2} + 141 a - 8\) , \( 2643 a^{3} - 5996 a^{2} - 2030 a + 1279\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(84a^{3}-267a^{2}+141a-8\right){x}+2643a^{3}-5996a^{2}-2030a+1279$ |
| 25.2-f4 |
25.2-f |
$4$ |
$4$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.606791681$ |
$719.1402936$ |
3.411940230 |
\( -\frac{632164}{25} a^{3} - \frac{867543}{5} a^{2} + \frac{12706826}{25} a + \frac{7522477}{25} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - a^{2} - 6 a + 2\) , \( a^{2} + a - 3\) , \( -a^{3} - 2 a^{2} + 6 a + 2\) , \( -3 a^{2} + 3 a + 3\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+\left(a^{2}+a-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a+2\right){x}^{2}+\left(-a^{3}-2a^{2}+6a+2\right){x}-3a^{2}+3a+3$ |
| 25.2-g1 |
25.2-g |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$74.91961994$ |
2.343169631 |
\( -\frac{716724786161}{390625} a^{3} + \frac{1442006043167}{390625} a^{2} + \frac{880238515792}{390625} a - \frac{24734039638}{390625} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 4\) , \( a^{3} - 5 a\) , \( 1\) , \( -9 a^{3} + 5 a^{2} + 61 a - 11\) , \( -14 a^{3} + 11 a^{2} + 91 a - 38\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-4\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(-9a^{3}+5a^{2}+61a-11\right){x}-14a^{3}+11a^{2}+91a-38$ |
| 25.2-g2 |
25.2-g |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$299.6784797$ |
2.343169631 |
\( \frac{194297}{625} a^{3} - \frac{281609}{625} a^{2} + \frac{263541}{625} a + \frac{1172301}{625} \) |
\( \bigl[a^{3} + a^{2} - 5 a - 4\) , \( a^{3} - 5 a\) , \( 1\) , \( a^{3} + a + 14\) , \( a^{3} + 2 a^{2} + 2 a + 6\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-4\right){x}{y}+{y}={x}^{3}+\left(a^{3}-5a\right){x}^{2}+\left(a^{3}+a+14\right){x}+a^{3}+2a^{2}+2a+6$ |
| 25.2-h1 |
25.2-h |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{12} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.057121782$ |
$542.2560016$ |
2.906267159 |
\( \frac{50670520594}{1953125} a^{3} - \frac{100307515486}{1953125} a^{2} - \frac{13618694406}{390625} a + \frac{8341092601}{1953125} \) |
\( \bigl[a^{2} + a - 3\) , \( a^{3} + a^{2} - 5 a - 3\) , \( a^{3} + a^{2} - 6 a - 3\) , \( 13 a^{3} + 9 a^{2} - 64 a - 34\) , \( -35 a^{3} - 2 a^{2} + 234 a + 113\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}+a^{2}-6a-3\right){y}={x}^{3}+\left(a^{3}+a^{2}-5a-3\right){x}^{2}+\left(13a^{3}+9a^{2}-64a-34\right){x}-35a^{3}-2a^{2}+234a+113$ |
| 25.2-h2 |
25.2-h |
$2$ |
$3$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.2 |
\( 5^{2} \) |
\( - 5^{4} \) |
$17.08964$ |
$(a^2+2a), (a^3-6a)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.171365346$ |
$542.2560016$ |
2.906267159 |
\( \frac{37687449}{125} a^{3} - \frac{19419931}{125} a^{2} - \frac{43245961}{25} a + \frac{73603996}{125} \) |
\( \bigl[a^{3} - 5 a - 1\) , \( -a^{3} + 6 a + 1\) , \( a^{3} - 5 a - 1\) , \( -16 a^{3} - 3 a^{2} + 89 a + 44\) , \( -28 a^{3} + 2 a^{2} + 142 a + 68\bigr] \) |
${y}^2+\left(a^{3}-5a-1\right){x}{y}+\left(a^{3}-5a-1\right){y}={x}^{3}+\left(-a^{3}+6a+1\right){x}^{2}+\left(-16a^{3}-3a^{2}+89a+44\right){x}-28a^{3}+2a^{2}+142a+68$ |
| 25.3-a1 |
25.3-a |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{8} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$70.73372866$ |
1.106126306 |
\( -\frac{196363213209817}{25} a^{3} + \frac{455662960639623}{25} a^{2} + \frac{24185133793363}{5} a - \frac{84578299065318}{25} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 7 a\) , \( a\) , \( 15 a^{3} - 37 a^{2} + 16 a - 40\) , \( 188 a^{3} - 382 a^{2} - 293 a + 187\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+7a\right){x}^{2}+\left(15a^{3}-37a^{2}+16a-40\right){x}+188a^{3}-382a^{2}-293a+187$ |
| 25.3-a2 |
25.3-a |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$141.4674573$ |
1.106126306 |
\( \frac{353613099}{625} a^{3} - \frac{838588956}{625} a^{2} - \frac{32093401}{125} a + \frac{136333371}{625} \) |
\( \bigl[a^{2} - 3\) , \( -a^{3} + 7 a\) , \( a\) , \( -2 a^{2} + 6 a\) , \( 4 a^{3} - 4 a^{2} - 14 a - 1\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}+7a\right){x}^{2}+\left(-2a^{2}+6a\right){x}+4a^{3}-4a^{2}-14a-1$ |
| 25.3-b1 |
25.3-b |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$72.97083102$ |
0.570554934 |
\( 47627 a^{3} + 17029 a^{2} - 280617 a - 146577 \) |
\( \bigl[a^{3} + a^{2} - 5 a - 3\) , \( -a^{3} + 5 a + 2\) , \( a^{2} - 3\) , \( 22 a^{3} + 9 a^{2} - 127 a - 65\) , \( 181 a^{3} + 63 a^{2} - 1061 a - 539\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-5a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(22a^{3}+9a^{2}-127a-65\right){x}+181a^{3}+63a^{2}-1061a-539$ |
| 25.3-c1 |
25.3-c |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{6} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$20.45424524$ |
0.319861248 |
\( 771693999106 a^{3} - 429571775989 a^{2} - 4420639064801 a + 1694388586445 \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 5 a + 5\) , \( a^{3} - 5 a\) , \( 11 a^{3} - 2 a^{2} - 72 a - 56\) , \( -47 a^{3} - 35 a^{2} + 263 a + 59\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+5\right){x}^{2}+\left(11a^{3}-2a^{2}-72a-56\right){x}-47a^{3}-35a^{2}+263a+59$ |
| 25.3-c2 |
25.3-c |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$40.90849049$ |
0.319861248 |
\( 295053642 a^{3} + 738500581 a^{2} + 73882199 a - 118866804 \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 5 a + 5\) , \( a^{3} - 5 a\) , \( -4 a^{3} - 7 a^{2} + 8 a + 9\) , \( -9 a^{3} - 19 a^{2} + 8 a + 5\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-5a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+5a+5\right){x}^{2}+\left(-4a^{3}-7a^{2}+8a+9\right){x}-9a^{3}-19a^{2}+8a+5$ |
| 25.3-d1 |
25.3-d |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{6} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$268.9117337$ |
4.205212256 |
\( 771693999106 a^{3} - 429571775989 a^{2} - 4420639064801 a + 1694388586445 \) |
\( \bigl[a^{2} + a - 3\) , \( -a^{3} - a^{2} + 6 a + 5\) , \( a^{3} - 6 a\) , \( -5 a^{3} - 58 a^{2} - 2 a + 18\) , \( -208 a^{3} - 211 a^{2} + 13 a + 32\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(-a^{3}-a^{2}+6a+5\right){x}^{2}+\left(-5a^{3}-58a^{2}-2a+18\right){x}-208a^{3}-211a^{2}+13a+32$ |
| 25.3-d2 |
25.3-d |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$537.8234674$ |
4.205212256 |
\( 295053642 a^{3} + 738500581 a^{2} + 73882199 a - 118866804 \) |
\( \bigl[a^{3} - 5 a\) , \( -a^{2} - a + 4\) , \( 1\) , \( 10 a^{3} + 2 a^{2} - 61 a - 27\) , \( -16 a^{3} - 6 a^{2} + 93 a + 48\bigr] \) |
${y}^2+\left(a^{3}-5a\right){x}{y}+{y}={x}^{3}+\left(-a^{2}-a+4\right){x}^{2}+\left(10a^{3}+2a^{2}-61a-27\right){x}-16a^{3}-6a^{2}+93a+48$ |
| 25.3-e1 |
25.3-e |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{6} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Ns |
$1$ |
\( 1 \) |
$1$ |
$227.4558869$ |
1.778465130 |
\( 47627 a^{3} + 17029 a^{2} - 280617 a - 146577 \) |
\( \bigl[a^{2} + a - 2\) , \( a + 1\) , \( a^{3} - 6 a\) , \( 6 a^{3} + 7 a^{2} - 22 a - 8\) , \( 10 a^{3} + 13 a^{2} - 33 a - 21\bigr] \) |
${y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}-6a\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a^{3}+7a^{2}-22a-8\right){x}+10a^{3}+13a^{2}-33a-21$ |
| 25.3-f1 |
25.3-f |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( 5^{10} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$205.3726611$ |
1.605797597 |
\( \frac{353613099}{625} a^{3} - \frac{838588956}{625} a^{2} - \frac{32093401}{125} a + \frac{136333371}{625} \) |
\( \bigl[a\) , \( a^{2} + a - 2\) , \( a^{2} - 3\) , \( 7 a^{3} - 14 a^{2} - 3 a + 3\) , \( -27 a^{3} + 65 a^{2} + 17 a - 14\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(7a^{3}-14a^{2}-3a+3\right){x}-27a^{3}+65a^{2}+17a-14$ |
| 25.3-f2 |
25.3-f |
$2$ |
$2$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
25.3 |
\( 5^{2} \) |
\( - 5^{8} \) |
$17.08964$ |
$(a^3-6a)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$102.6863305$ |
1.605797597 |
\( -\frac{196363213209817}{25} a^{3} + \frac{455662960639623}{25} a^{2} + \frac{24185133793363}{5} a - \frac{84578299065318}{25} \) |
\( \bigl[a\) , \( a^{2} + a - 2\) , \( a^{2} - 3\) , \( 107 a^{3} - 249 a^{2} - 53 a + 38\) , \( -2023 a^{3} + 4688 a^{2} + 1270 a - 874\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}+a-2\right){x}^{2}+\left(107a^{3}-249a^{2}-53a+38\right){x}-2023a^{3}+4688a^{2}+1270a-874$ |
| 27.1-a1 |
27.1-a |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{6} \) |
$17.25484$ |
$(-a^3+a^2+5a-4)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.068314291$ |
$1099.868286$ |
4.699919391 |
\( \frac{2253977}{3} a^{3} + \frac{1315172}{9} a^{2} - \frac{37915672}{9} a - \frac{6303316}{3} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 3\) , \( a^{3} - 7 a\) , \( a\) , \( -2 a^{3} - a^{2} + 3\) , \( -4 a^{3} - 8 a^{2} - 2 a + 2\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-3\right){x}{y}+a{y}={x}^{3}+\left(a^{3}-7a\right){x}^{2}+\left(-2a^{3}-a^{2}+3\right){x}-4a^{3}-8a^{2}-2a+2$ |
| 27.1-b1 |
27.1-b |
$1$ |
$1$ |
4.4.16357.1 |
$4$ |
$[4, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{6} \) |
$17.25484$ |
$(-a^3+a^2+5a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$119.0724684$ |
1.862042227 |
\( \frac{2253977}{3} a^{3} + \frac{1315172}{9} a^{2} - \frac{37915672}{9} a - \frac{6303316}{3} \) |
\( \bigl[a^{3} + a^{2} - 6 a - 4\) , \( -a^{3} - a^{2} + 5 a + 4\) , \( 1\) , \( -20 a^{3} - 20 a^{2} + 81 a + 45\) , \( -62 a^{3} - 109 a^{2} + 115 a + 79\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-6a-4\right){x}{y}+{y}={x}^{3}+\left(-a^{3}-a^{2}+5a+4\right){x}^{2}+\left(-20a^{3}-20a^{2}+81a+45\right){x}-62a^{3}-109a^{2}+115a+79$ |
*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.
