| 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 |
| 21.1-a1 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.128168662$ |
$87.10503032$ |
3.724468981 |
\( \frac{283422372242314}{17294403} a^{3} + \frac{566742902699713}{17294403} a^{2} - \frac{11226391012049}{5764801} a - \frac{165429125280241}{17294403} \) |
\( \bigl[a^{3} - 5 a - 2\) , \( a^{2} - 3\) , \( a\) , \( 72 a^{3} + 44 a^{2} - 417 a - 462\) , \( 786 a^{3} + 381 a^{2} - 4500 a - 4563\bigr] \) |
${y}^2+\left(a^{3}-5a-2\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(72a^{3}+44a^{2}-417a-462\right){x}+786a^{3}+381a^{2}-4500a-4563$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$4.256337325$ |
$87.10503032$ |
3.724468981 |
\( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -13750 a^{3} + 16600 a^{2} + 62447 a - 34158\) , \( 1313070 a^{3} - 1585569 a^{2} - 5963840 a + 3262240\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16600a^{2}+62447a-34158\right){x}+1313070a^{3}-1585569a^{2}-5963840a+3262240$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{32} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$8.512674650$ |
$5.444064395$ |
3.724468981 |
\( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -13750 a^{3} + 16605 a^{2} + 62452 a - 34158\) , \( 1313123 a^{3} - 1585648 a^{2} - 5964098 a + 3262387\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-13750a^{3}+16605a^{2}+62452a-34158\right){x}+1313123a^{3}-1585648a^{2}-5964098a+3262387$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$4.256337325$ |
$5.444064395$ |
3.724468981 |
\( \frac{48823353105681873974}{7203} a^{3} + \frac{126481197886698394861}{7203} a^{2} + \frac{11571783696151730607}{2401} a - \frac{18846846758411390426}{2401} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -930 a^{3} + 1070 a^{2} + 4157 a - 2268\) , \( 17576 a^{3} - 21717 a^{2} - 80504 a + 44128\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-930a^{3}+1070a^{2}+4157a-2268\right){x}+17576a^{3}-21717a^{2}-80504a+44128$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.064084331$ |
$87.10503032$ |
3.724468981 |
\( -\frac{25780022}{21609} a^{3} - \frac{18946955}{21609} a^{2} + \frac{5547721}{2401} a - \frac{16556678}{21609} \) |
\( \bigl[1\) , \( a\) , \( a^{2} - 2\) , \( -50 a^{3} + 60 a^{2} + 227 a - 123\) , \( 382 a^{3} - 462 a^{2} - 1736 a + 949\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-2\right){y}={x}^{3}+a{x}^{2}+\left(-50a^{3}+60a^{2}+227a-123\right){x}+382a^{3}-462a^{2}-1736a+949$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.128168662$ |
$87.10503032$ |
3.724468981 |
\( \frac{73204218707003465754418530011323778}{17294403} a^{3} - \frac{29465194712745950920219395499999665}{5764801} a^{2} - \frac{110828615043320066217968092545904435}{5764801} a + \frac{60623589840441817561364743725976875}{5764801} \) |
\( \bigl[a + 1\) , \( a^{3} - a^{2} - 5 a\) , \( a^{2} - 3\) , \( -784 a^{3} - 509 a^{2} + 1519 a - 616\) , \( 18547 a^{3} + 24195 a^{2} - 19665 a + 1670\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-784a^{3}-509a^{2}+1519a-616\right){x}+18547a^{3}+24195a^{2}-19665a+1670$ |
| 21.1-b1 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.661230424$ |
$168.9045292$ |
2.243929483 |
\( \frac{2038984884625177049}{189} a^{3} - \frac{7386347250266428568}{567} a^{2} - \frac{9260854421312515172}{189} a + \frac{15197146687274038624}{567} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 54 a^{3} + 37 a^{2} - 318 a - 386\) , \( -269 a^{3} - 197 a^{2} + 1611 a + 1838\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(54a^{3}+37a^{2}-318a-386\right){x}-269a^{3}-197a^{2}+1611a+1838$ |
| 21.1-b2 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{2} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.322460848$ |
$168.9045292$ |
2.243929483 |
\( -\frac{2514435768239}{147} a^{3} + \frac{14274453830897}{441} a^{2} + \frac{2028661440026}{49} a - \frac{11951649907706}{441} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 34 a^{3} + 12 a^{2} - 188 a - 186\) , \( 183 a^{3} + 62 a^{2} - 1009 a - 987\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(34a^{3}+12a^{2}-188a-186\right){x}+183a^{3}+62a^{2}-1009a-987$ |
| 21.1-b3 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.661230424$ |
$168.9045292$ |
2.243929483 |
\( \frac{108278519}{2401} a^{3} - \frac{629396917}{7203} a^{2} - \frac{246620209}{2401} a + \frac{494923985}{7203} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + 2 a^{2} + 2 a - 1\) , \( 9 a^{3} + 3 a^{2} - 50 a - 49\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+1\right){y}={x}^{3}+\left(a^{3}-5a-2\right){x}^{2}+\left(-a^{3}+2a^{2}+2a-1\right){x}+9a^{3}+3a^{2}-50a-49$ |
| 21.1-b4 |
21.1-b |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$2.644921697$ |
$10.55653307$ |
2.243929483 |
\( -\frac{15112197778035174188315}{7} a^{3} + \frac{85702869193556526085960}{21} a^{2} + \frac{36669851114797762306924}{7} a - \frac{71948817390117157608464}{21} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 3 a + 1\) , \( 0\) , \( -90 a^{3} + 80 a^{2} + 433 a - 231\) , \( 813 a^{3} - 1180 a^{2} - 3693 a + 2070\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}={x}^{3}+\left(a^{3}-a^{2}-3a+1\right){x}^{2}+\left(-90a^{3}+80a^{2}+433a-231\right){x}+813a^{3}-1180a^{2}-3693a+2070$ |
| 21.1-c1 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{16} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$2.049678336$ |
$17.69737756$ |
2.915209027 |
\( \frac{2038984884625177049}{189} a^{3} - \frac{7386347250266428568}{567} a^{2} - \frac{9260854421312515172}{189} a + \frac{15197146687274038624}{567} \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{3} - 5 a - 2\) , \( 247 a^{3} - 369 a^{2} - 783 a + 29\) , \( -2748 a^{3} + 5314 a^{2} + 6231 a - 5064\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}-5a-2\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(247a^{3}-369a^{2}-783a+29\right){x}-2748a^{3}+5314a^{2}+6231a-5064$ |
| 21.1-c2 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{2} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.024839168$ |
$283.1580409$ |
2.915209027 |
\( -\frac{2514435768239}{147} a^{3} + \frac{14274453830897}{441} a^{2} + \frac{2028661440026}{49} a - \frac{11951649907706}{441} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a + 1\) , \( 203 a^{3} + 99 a^{2} - 1170 a - 1195\) , \( -3273 a^{3} - 1665 a^{2} + 18781 a + 19343\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(203a^{3}+99a^{2}-1170a-1195\right){x}-3273a^{3}-1665a^{2}+18781a+19343$ |
| 21.1-c3 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.512419584$ |
$283.1580409$ |
2.915209027 |
\( \frac{108278519}{2401} a^{3} - \frac{629396917}{7203} a^{2} - \frac{246620209}{2401} a + \frac{494923985}{7203} \) |
\( \bigl[a^{3} - a^{2} - 4 a + 1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a + 1\) , \( 3 a^{3} - a^{2} - 15 a - 5\) , \( -135 a^{3} - 70 a^{2} + 775 a + 803\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(3a^{3}-a^{2}-15a-5\right){x}-135a^{3}-70a^{2}+775a+803$ |
| 21.1-c4 |
21.1-c |
$4$ |
$4$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7 \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.512419584$ |
$283.1580409$ |
2.915209027 |
\( -\frac{15112197778035174188315}{7} a^{3} + \frac{85702869193556526085960}{21} a^{2} + \frac{36669851114797762306924}{7} a - \frac{71948817390117157608464}{21} \) |
\( \bigl[a + 1\) , \( a^{3} - 4 a - 2\) , \( a^{2} - 2\) , \( -167 a^{3} - 178 a^{2} + 229 a - 60\) , \( 1329 a^{3} + 5176 a^{2} + 3359 a - 3168\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(a^{3}-4a-2\right){x}^{2}+\left(-167a^{3}-178a^{2}+229a-60\right){x}+1329a^{3}+5176a^{2}+3359a-3168$ |
| 21.1-d1 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1.654559453$ |
$353.3517645$ |
2.936599577 |
\( \frac{283422372242314}{17294403} a^{3} + \frac{566742902699713}{17294403} a^{2} - \frac{11226391012049}{5764801} a - \frac{165429125280241}{17294403} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{3} - 5 a - 3\) , \( a^{3} - a^{2} - 3 a + 2\) , \( 87 a^{3} - 17 a^{2} - 418 a - 327\) , \( -740 a^{3} - 97 a^{2} + 3864 a + 3503\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{3}-a^{2}-3a+2\right){y}={x}^{3}+\left(a^{3}-5a-3\right){x}^{2}+\left(87a^{3}-17a^{2}-418a-327\right){x}-740a^{3}-97a^{2}+3864a+3503$ |
| 21.1-d2 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{8} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$6.618237813$ |
$1.380280330$ |
2.936599577 |
\( \frac{73204218707003465754418530011323778}{17294403} a^{3} - \frac{29465194712745950920219395499999665}{5764801} a^{2} - \frac{110828615043320066217968092545904435}{5764801} a + \frac{60623589840441817561364743725976875}{5764801} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -544 a^{3} + 506 a^{2} + 2629 a - 1404\) , \( -13092 a^{3} + 14266 a^{2} + 61674 a - 33259\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-544a^{3}+506a^{2}+2629a-1404\right){x}-13092a^{3}+14266a^{2}+61674a-33259$ |
| 21.1-d3 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{32} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$6.618237813$ |
$1.380280330$ |
2.936599577 |
\( -\frac{8726799587448504883954432291309058}{3313283022731761938915897603} a^{3} + \frac{3512609295216958802166282509619697}{1104427674243920646305299201} a^{2} + \frac{13211649577996599292098856645855219}{1104427674243920646305299201} a - \frac{7226027092698629340082787287567899}{1104427674243920646305299201} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -84 a^{3} + 146 a^{2} + 299 a - 174\) , \( -188 a^{3} + 858 a^{2} + 1498 a - 961\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-84a^{3}+146a^{2}+299a-174\right){x}-188a^{3}+858a^{2}+1498a-961$ |
| 21.1-d4 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$3.309118906$ |
$22.08448528$ |
2.936599577 |
\( \frac{3075730896093395126238965002}{99698791708803} a^{3} - \frac{3714007662409537173380188813}{99698791708803} a^{2} - \frac{4656548508714567651606714255}{33232930569601} a + \frac{2547146210989604914629326570}{33232930569601} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( -34 a^{3} + 6 a^{2} + 144 a - 69\) , \( -240 a^{3} + 410 a^{2} + 1244 a - 706\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-34a^{3}+6a^{2}+144a-69\right){x}-240a^{3}+410a^{2}+1244a-706$ |
| 21.1-d5 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.827279726$ |
$353.3517645$ |
2.936599577 |
\( -\frac{25780022}{21609} a^{3} - \frac{18946955}{21609} a^{2} + \frac{5547721}{2401} a - \frac{16556678}{21609} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( a^{3} - 4 a^{2} - 6 a + 6\) , \( a^{2} - a - 1\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(a^{3}-4a^{2}-6a+6\right){x}+a^{2}-a-1$ |
| 21.1-d6 |
21.1-d |
$6$ |
$8$ |
4.4.9909.1 |
$4$ |
$[4, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{4} \) |
$13.01460$ |
$(-a^3+a^2+4a), (a^3-a^2-4a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.827279726$ |
$353.3517645$ |
2.936599577 |
\( \frac{48823353105681873974}{7203} a^{3} + \frac{126481197886698394861}{7203} a^{2} + \frac{11571783696151730607}{2401} a - \frac{18846846758411390426}{2401} \) |
\( \bigl[a^{3} - a^{2} - 3 a + 2\) , \( a^{3} - a^{2} - 5 a\) , \( a + 1\) , \( 36 a^{3} - 554 a^{2} - 426 a + 351\) , \( 660 a^{3} + 9720 a^{2} + 6098 a - 5674\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-3a+2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(36a^{3}-554a^{2}-426a+351\right){x}+660a^{3}+9720a^{2}+6098a-5674$ |
*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.
