| 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 |
| 567.2-a1 |
567.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( 3^{18} \cdot 7^{2} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.993662842$ |
1.112995248 |
\( \frac{290371}{441} a - \frac{71584}{189} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 21 a + 45\) , \( 39 a + 86\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(21a+45\right){x}+39a+86$ |
| 567.2-a2 |
567.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( 3^{24} \cdot 7 \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.993662842$ |
1.112995248 |
\( -\frac{1959349475}{1701} a + \frac{2682142237}{729} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -69 a - 180\) , \( -114 a - 202\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-69a-180\right){x}-114a-202$ |
| 567.2-b1 |
567.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{12} \cdot 7^{2} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.981195039$ |
0.367898682 |
\( -\frac{213433415640625}{49} a + \frac{97343395248346}{7} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -8 a - 266\) , \( 673 a + 23\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-8a-266\right){x}+673a+23$ |
| 567.2-b2 |
567.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{12} \cdot 7^{10} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.981195039$ |
0.367898682 |
\( -\frac{91176666325}{282475249} a + \frac{41583934921}{40353607} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 34 a - 113\) , \( 8 a - 57\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(34a-113\right){x}+8a-57$ |
| 567.2-b3 |
567.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{12} \cdot 7^{5} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2 \) |
$1$ |
$3.962390079$ |
0.367898682 |
\( \frac{173650213}{16807} a + \frac{58516320}{2401} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -11 a + 22\) , \( -10 a + 24\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-11a+22\right){x}-10a+24$ |
| 567.2-b4 |
567.2-b |
$4$ |
$10$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{12} \cdot 7 \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2 \) |
$1$ |
$3.962390079$ |
0.367898682 |
\( \frac{94831363}{7} a + 24861195 \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -53 a - 131\) , \( 385 a + 824\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-53a-131\right){x}+385a+824$ |
| 567.2-c1 |
567.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{6} \cdot 7^{5} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$11.71788562$ |
4.351913466 |
\( -\frac{4091904}{16807} a + \frac{2875392}{2401} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 3 a + 6\) , \( 7 a + 15\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(3a+6\right){x}+7a+15$ |
| 567.2-d1 |
567.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{14} \cdot 7 \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$6.565193697$ |
2.743033193 |
\( -\frac{7309}{21} a + 984 \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 24 a - 77\) , \( -17 a + 54\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(24a-77\right){x}-17a+54$ |
| 567.2-d2 |
567.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( 3^{20} \cdot 7 \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$9$ |
\( 2^{2} \) |
$1$ |
$1.641298424$ |
2.743033193 |
\( -\frac{1452729479479}{189} a + \frac{1987696672010}{81} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -324 a - 738\) , \( -157 a - 429\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-324a-738\right){x}-157a-429$ |
| 567.2-d3 |
567.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( 3^{16} \cdot 7^{2} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$9$ |
\( 2^{3} \) |
$1$ |
$3.282596848$ |
2.743033193 |
\( \frac{12647125}{441} a + \frac{16825789}{63} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -234 a - 513\) , \( -3064 a - 6720\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-234a-513\right){x}-3064a-6720$ |
| 567.2-d4 |
567.2-d |
$4$ |
$4$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{14} \cdot 7^{4} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$9$ |
\( 2^{3} \) |
$1$ |
$0.820649212$ |
2.743033193 |
\( \frac{171321456375145}{7203} a + \frac{17887448107818}{343} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 114 a - 392\) , \( 7111 a - 22752\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(114a-392\right){x}+7111a-22752$ |
| 567.2-e1 |
567.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{18} \cdot 7^{5} \) |
$2.34819$ |
$(a-1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.864993246$ |
1.064031779 |
\( -\frac{4091904}{16807} a + \frac{2875392}{2401} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 27 a + 54\) , \( -210 a - 461\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(27a+54\right){x}-210a-461$ |
| 567.2-f1 |
567.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{14} \cdot 7^{9} \) |
$2.34819$ |
$(a-1), (3)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3^{2} \) |
$1.119975381$ |
$4.802101195$ |
3.994852757 |
\( -\frac{178100973154304}{121060821} a - \frac{15798789210112}{5764801} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 1143 a - 3711\) , \( -34941 a + 111735\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(1143a-3711\right){x}-34941a+111735$ |
| 567.2-f2 |
567.2-f |
$2$ |
$3$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
567.2 |
\( 3^{4} \cdot 7 \) |
\( - 3^{18} \cdot 7^{3} \) |
$2.34819$ |
$(a-1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$0.373325127$ |
$4.802101195$ |
3.994852757 |
\( -\frac{15929344}{9261} a + \frac{7266304}{1323} \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 63 a - 201\) , \( 402 a - 1287\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(63a-201\right){x}+402a-1287$ |
*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.
