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 |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$1.20695$ |
$(-a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$29.18343844$ |
2.569458482 |
\( -\frac{958464783206251}{2} a + \frac{5922266046113995}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( -117348739 a - 607738365\) , \( 1649856442368 a + 8544455298751\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-117348739a-607738365\right){x}+1649856442368a+8544455298751$ |
2.1-a2 |
2.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$1.20695$ |
$(-a+6)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$29.18343844$ |
2.569458482 |
\( -\frac{2651}{32} a + \frac{59835}{32} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 1311181 a + 6790485\) , \( -1010575092 a - 5233675779\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+{x}^{2}+\left(1311181a+6790485\right){x}-1010575092a-5233675779$ |
2.1-b1 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( -2 \) |
$1.20695$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$5.203755382$ |
$1.150939947$ |
1.054641065 |
\( -\frac{958464783206251}{2} a + \frac{5922266046113995}{2} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 731 a - 4419\) , \( 24963 a - 154005\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(731a-4419\right){x}+24963a-154005$ |
2.1-b2 |
2.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.1 |
\( 2 \) |
\( - 2^{5} \) |
$1.20695$ |
$(-a+6)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1.040751076$ |
$28.77349869$ |
1.054641065 |
\( -\frac{2651}{32} a + \frac{59835}{32} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 11 a + 31\) , \( 23 a + 105\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(11a+31\right){x}+23a+105$ |
2.2-a1 |
2.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{5} \) |
$1.20695$ |
$(a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$29.18343844$ |
2.569458482 |
\( \frac{2651}{32} a + 1787 \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( -1311181 a + 8101666\) , \( 1011886273 a - 6252352537\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1311181a+8101666\right){x}+1011886273a-6252352537$ |
2.2-a2 |
2.2-a |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( -2 \) |
$1.20695$ |
$(a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$29.18343844$ |
2.569458482 |
\( \frac{958464783206251}{2} a + 2481900631453872 \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 117348739 a - 725087104\) , \( -1649973791107 a + 10195036828223\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(117348739a-725087104\right){x}-1649973791107a+10195036828223$ |
2.2-b1 |
2.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( - 2^{5} \) |
$1.20695$ |
$(a+5)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1.040751076$ |
$28.77349869$ |
1.054641065 |
\( \frac{2651}{32} a + 1787 \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( 4 a + 10\) , \( 2 a + 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(4a+10\right){x}+2a+6$ |
2.2-b2 |
2.2-b |
$2$ |
$5$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
2.2 |
\( 2 \) |
\( -2 \) |
$1.20695$ |
$(a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$5.203755382$ |
$1.150939947$ |
1.054641065 |
\( \frac{958464783206251}{2} a + 2481900631453872 \) |
\( \bigl[a\) , \( a - 1\) , \( a + 1\) , \( -716 a - 3720\) , \( -28668 a - 148474\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-716a-3720\right){x}-28668a-148474$ |
4.1-a1 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.151409674$ |
4.386134892 |
\( -\frac{60716385}{64} a - \frac{19652787}{4} \) |
\( \bigl[1\) , \( a\) , \( a\) , \( -80 a + 505\) , \( 308288 a - 1904883\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-80a+505\right){x}+308288a-1904883$ |
4.1-a2 |
4.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$4.151409674$ |
4.386134892 |
\( \frac{60716385}{64} a - \frac{375160977}{64} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( 79 a + 425\) , \( -308289 a - 1596595\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(79a+425\right){x}-308289a-1596595$ |
4.1-b1 |
4.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6), (a+5)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.022799893$ |
$36.76371340$ |
1.180802658 |
\( -\frac{60716385}{64} a - \frac{19652787}{4} \) |
\( \bigl[1\) , \( -a + 1\) , \( a\) , \( -16776 a - 86865\) , \( 2846099 a + 14739689\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-16776a-86865\right){x}+2846099a+14739689$ |
4.1-b2 |
4.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6), (a+5)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2^{2} \) |
$0.022799893$ |
$36.76371340$ |
1.180802658 |
\( \frac{60716385}{64} a - \frac{375160977}{64} \) |
\( \bigl[1\) , \( a\) , \( a + 1\) , \( 16775 a - 103641\) , \( -2846100 a + 17585788\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(16775a-103641\right){x}-2846100a+17585788$ |
4.2-a1 |
4.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$1.337287870$ |
$4.214438833$ |
2.977289430 |
\( -6164480 a - 31924224 \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( -393024 a + 2428472\) , \( 255241909 a - 1577116380\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(-393024a+2428472\right){x}+255241909a-1577116380$ |
4.2-b1 |
4.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$43$ |
43Ns.9.1 |
$1$ |
\( 1 \) |
$0.597796035$ |
$17.69503190$ |
1.862685445 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( a + 11\) , \( -12145 a + 75047\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+11\right){x}-12145a+75047$ |
4.2-b2 |
4.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$43$ |
43Ns.9.1 |
$1$ |
\( 3 \) |
$0.199265345$ |
$17.69503190$ |
1.862685445 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( a + 11\) , \( 905486 a + 4689425\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+11\right){x}+905486a+4689425$ |
4.2-c1 |
4.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(-a+6)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.309179342$ |
$29.09037348$ |
1.583780192 |
\( -6164480 a - 31924224 \) |
\( \bigl[0\) , \( -1\) , \( a\) , \( -259 a - 1341\) , \( 5493 a + 28441\bigr] \) |
${y}^2+a{y}={x}^{3}-{x}^{2}+\left(-259a-1341\right){x}+5493a+28441$ |
4.3-a1 |
4.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 3 \) |
$1.337287870$ |
$4.214438833$ |
2.977289430 |
\( 6164480 a - 38088704 \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( 393024 a + 2035448\) , \( -255241910 a - 1321874471\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(393024a+2035448\right){x}-255241910a-1321874471$ |
4.3-b1 |
4.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$43$ |
43Ns.9.1 |
$1$ |
\( 1 \) |
$0.597796035$ |
$17.69503190$ |
1.862685445 |
\( 0 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a + 11\) , \( 12144 a + 62891\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+11\right){x}+12144a+62891$ |
4.3-b2 |
4.3-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$43$ |
43Ns.9.1 |
$1$ |
\( 3 \) |
$0.199265345$ |
$17.69503190$ |
1.862685445 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 11\) , \( -905487 a + 5594922\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+11\right){x}-905487a+5594922$ |
4.3-c1 |
4.3-c |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
4.3 |
\( 2^{2} \) |
\( 2^{8} \) |
$1.43532$ |
$(a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.309179342$ |
$29.09037348$ |
1.583780192 |
\( 6164480 a - 38088704 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 259 a - 1600\) , \( -5494 a + 33934\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(259a-1600\right){x}-5494a+33934$ |
6.1-a1 |
6.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$1.58844$ |
$(a+5), (-28a+173)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.458701152$ |
$5.611992640$ |
3.626369488 |
\( \frac{7198885}{432} a - \frac{911579}{9} \) |
\( \bigl[1\) , \( 1\) , \( a\) , \( 1070 a - 6614\) , \( 46088 a - 284783\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(1070a-6614\right){x}+46088a-284783$ |
6.1-b1 |
6.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
6.1 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$1.58844$ |
$(a+5), (-28a+173)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \) |
$0.032203325$ |
$20.75126785$ |
1.412087948 |
\( \frac{7198885}{432} a - \frac{911579}{9} \) |
\( \bigl[1\) , \( a + 1\) , \( 1\) , \( 55114 a + 285435\) , \( -2721536 a - 14094588\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(55114a+285435\right){x}-2721536a-14094588$ |
6.2-a1 |
6.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$1.58844$ |
$(-a+6), (-28a+173)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{3} \) |
$0.458701152$ |
$5.611992640$ |
3.626369488 |
\( -\frac{7198885}{432} a - \frac{36556907}{432} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -1071 a - 5544\) , \( -46089 a - 238695\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-1071a-5544\right){x}-46089a-238695$ |
6.2-b1 |
6.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
6.2 |
\( 2 \cdot 3 \) |
\( 2^{4} \cdot 3^{6} \) |
$1.58844$ |
$(-a+6), (-28a+173)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2^{2} \cdot 3 \) |
$0.032203325$ |
$20.75126785$ |
1.412087948 |
\( -\frac{7198885}{432} a - \frac{36556907}{432} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -55112 a + 340548\) , \( 2776649 a - 17156672\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-55112a+340548\right){x}+2776649a-17156672$ |
8.1-a1 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{38} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$4.348459975$ |
3.828605526 |
\( -\frac{172032790985}{1073741824} a + \frac{30074625879}{33554432} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -2580396 a - 13363628\) , \( 23701425877 a + 122747512292\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-2580396a-13363628\right){x}+23701425877a+122747512292$ |
8.1-a2 |
8.1-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.348459975$ |
3.828605526 |
\( \frac{1157415}{1024} a - \frac{220601}{32} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 284339 a + 1472572\) , \( -837230821 a - 4335941676\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(284339a+1472572\right){x}-837230821a-4335941676$ |
8.1-b1 |
8.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{9} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.078509084$ |
$15.94185712$ |
1.944933359 |
\( \frac{2307}{2} a + 2608 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -973 a - 5013\) , \( 34727 a + 179889\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-973a-5013\right){x}+34727a+179889$ |
8.1-b2 |
8.1-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.235527253$ |
$1.771317458$ |
1.944933359 |
\( \frac{47232606203}{8} a - 36480294292 \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -17378 a - 89973\) , \( -2978637 a - 15426047\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-17378a-89973\right){x}-2978637a-15426047$ |
8.1-c1 |
8.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{9} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$29.10944917$ |
2.562944090 |
\( \frac{2307}{2} a + 2608 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 10100 a - 62389\) , \( -1593582 a + 9846631\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(10100a-62389\right){x}-1593582a+9846631$ |
8.1-c2 |
8.1-c |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$29.10944917$ |
2.562944090 |
\( \frac{47232606203}{8} a - 36480294292 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 869255 a - 5371029\) , \( -1049040548 a + 6481925431\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(869255a-5371029\right){x}-1049040548a+6481925431$ |
8.1-d1 |
8.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{38} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.911621126$ |
$2.188951422$ |
1.473679633 |
\( -\frac{172032790985}{1073741824} a + \frac{30074625879}{33554432} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -524 a + 3249\) , \( -1696 a + 10502\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-524a+3249\right){x}-1696a+10502$ |
8.1-d2 |
8.1-d |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.637207042$ |
$19.70056279$ |
1.473679633 |
\( \frac{1157415}{1024} a - \frac{220601}{32} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 91 a - 551\) , \( -980 a + 6078\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(91a-551\right){x}-980a+6078$ |
8.2-a1 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.348459975$ |
3.828605526 |
\( -\frac{1157415}{1024} a - \frac{5901817}{1024} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -284341 a + 1756911\) , \( 837230820 a - 5173172497\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-284341a+1756911\right){x}+837230820a-5173172497$ |
8.2-a2 |
8.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{38} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \cdot 5 \) |
$1$ |
$4.348459975$ |
3.828605526 |
\( \frac{172032790985}{1073741824} a + \frac{790355237143}{1073741824} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 2580394 a - 15944024\) , \( -23701425878 a + 146448938169\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(2580394a-15944024\right){x}-23701425878a+146448938169$ |
8.2-b1 |
8.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$6.235527253$ |
$1.771317458$ |
1.944933359 |
\( -\frac{47232606203}{8} a - \frac{244609748133}{8} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 17379 a - 107319\) , \( 2996015 a - 18512003\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(17379a-107319\right){x}+2996015a-18512003$ |
8.2-b2 |
8.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{9} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$2.078509084$ |
$15.94185712$ |
1.944933359 |
\( -\frac{2307}{2} a + \frac{7523}{2} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 974 a - 5954\) , \( -33754 a + 208662\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(974a-5954\right){x}-33754a+208662$ |
8.2-c1 |
8.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{11} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$29.10944917$ |
2.562944090 |
\( -\frac{47232606203}{8} a - \frac{244609748133}{8} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -869257 a - 4501774\) , \( 1049040547 a + 5432884883\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-869257a-4501774\right){x}+1049040547a+5432884883$ |
8.2-c2 |
8.2-c |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{9} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$29.10944917$ |
2.562944090 |
\( -\frac{2307}{2} a + \frac{7523}{2} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -10102 a - 52289\) , \( 1593581 a + 8253049\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-10102a-52289\right){x}+1593581a+8253049$ |
8.2-d1 |
8.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{18} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.637207042$ |
$19.70056279$ |
1.473679633 |
\( -\frac{1157415}{1024} a - \frac{5901817}{1024} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -93 a - 460\) , \( 979 a + 5098\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-93a-460\right){x}+979a+5098$ |
8.2-d2 |
8.2-d |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{38} \) |
$1.70689$ |
$(-a+6), (a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1.911621126$ |
$2.188951422$ |
1.473679633 |
\( \frac{172032790985}{1073741824} a + \frac{790355237143}{1073741824} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 522 a + 2725\) , \( 1695 a + 8806\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(522a+2725\right){x}+1695a+8806$ |
10.2-a1 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{6} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$9.391352793$ |
4.961175046 |
\( \frac{25965603}{31250} a - \frac{52921296}{15625} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 102948240 a - 636107660\) , \( 1499528208294 a - 9265447360847\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(102948240a-636107660\right){x}+1499528208294a-9265447360847$ |
10.2-a2 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{3} \cdot 5^{2} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$9.391352793$ |
4.961175046 |
\( \frac{1616685723}{200} a + \frac{1046625516}{25} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( -a + 9\) , \( -6 a - 23\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-a+9\right){x}-6a-23$ |
10.2-b1 |
10.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{2} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$17.47376826$ |
0.769239755 |
\( -\frac{31939}{50} a + \frac{98673}{25} \) |
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -9 a - 44\) , \( -6443 a - 33372\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-9a-44\right){x}-6443a-33372$ |
10.2-b2 |
10.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5 \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$17.47376826$ |
0.769239755 |
\( \frac{106969}{20} a + \frac{139871}{5} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 10\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-a{x}^{2}+10{x}-a+1$ |
10.2-c1 |
10.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{13} \cdot 5^{2} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1.124606412$ |
$5.853140175$ |
2.318219832 |
\( \frac{6711461587}{204800} a + \frac{1085736051}{6400} \) |
\( \bigl[a\) , \( 0\) , \( 1\) , \( 82977 a - 512680\) , \( 156511178 a - 967068192\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(82977a-512680\right){x}+156511178a-967068192$ |
10.2-d1 |
10.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{13} \cdot 5^{2} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 13 \) |
$0.023688676$ |
$23.63894871$ |
2.563760464 |
\( \frac{6711461587}{204800} a + \frac{1085736051}{6400} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( -807 a - 4157\) , \( 26887 a + 139262\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(-807a-4157\right){x}+26887a+139262$ |
10.2-e1 |
10.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{2} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$42.75671159$ |
1.882259273 |
\( -\frac{31939}{50} a + \frac{98673}{25} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 11246 a - 69472\) , \( -1333517 a + 8239667\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(11246a-69472\right){x}-1333517a+8239667$ |
10.2-e2 |
10.2-e |
$2$ |
$2$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5 \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$42.75671159$ |
1.882259273 |
\( \frac{106969}{20} a + \frac{139871}{5} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( -17574147 a + 108589040\) , \( -219640655600 a + 1357139479951\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-17574147a+108589040\right){x}-219640655600a+1357139479951$ |
10.2-f1 |
10.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2 \cdot 5^{6} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.067772993$ |
$26.27860898$ |
1.254452365 |
\( \frac{25965603}{31250} a - \frac{52921296}{15625} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -1\) , \( 1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}-{x}+1$ |
10.2-f2 |
10.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{129}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( - 2^{3} \cdot 5^{2} \) |
$1.80482$ |
$(a+5), (-6a+37)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.067772993$ |
$26.27860898$ |
1.254452365 |
\( \frac{1616685723}{200} a + \frac{1046625516}{25} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -73978694 a + 457107630\) , \( 132250044861 a - 817160905793\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-73978694a+457107630\right){x}+132250044861a-817160905793$ |
*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.