| 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 |
| 343.4-a1 |
343.4-a |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( - 7^{8} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$207.6574291$ |
3.202318121 |
\( \frac{40225241338678555}{49} a^{3} + \frac{74036012651745535}{49} a^{2} + \frac{9175766779869835}{49} a - \frac{14161143749601521}{49} \) |
\( \bigl[a^{2} - 2 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( 84 a^{3} - 136 a^{2} - 262 a - 89\) , \( -444 a^{3} + 693 a^{2} + 1433 a + 512\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+\left(-a^{3}+2a^{2}+4a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(84a^{3}-136a^{2}-262a-89\right){x}-444a^{3}+693a^{2}+1433a+512$ |
| 343.4-a2 |
343.4-a |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( - 7^{7} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$415.3148583$ |
3.202318121 |
\( \frac{2677817}{7} a^{3} - \frac{117781460}{7} a^{2} - \frac{145844186}{7} a + \frac{65824690}{7} \) |
\( \bigl[a^{2} - 2 a - 1\) , \( a^{3} - a^{2} - 5 a\) , \( -a^{3} + 2 a^{2} + 4 a - 1\) , \( -a^{3} - a^{2} + 8 a + 6\) , \( -17 a^{3} + 18 a^{2} + 74 a + 31\bigr] \) |
${y}^2+\left(a^{2}-2a-1\right){x}{y}+\left(-a^{3}+2a^{2}+4a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-5a\right){x}^{2}+\left(-a^{3}-a^{2}+8a+6\right){x}-17a^{3}+18a^{2}+74a+31$ |
| 343.4-a3 |
343.4-a |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( - 7^{11} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$83.06297166$ |
3.202318121 |
\( \frac{883009024}{16807} a^{3} - \frac{1375719561}{16807} a^{2} - \frac{3570496060}{16807} a + \frac{1021377428}{16807} \) |
\( \bigl[-a^{3} + 2 a^{2} + 4 a - 1\) , \( a^{3} - 2 a^{2} - 4 a\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( 5 a^{3} - 8 a^{2} - 18 a - 6\) , \( 9 a^{3} - 18 a^{2} - 22 a - 4\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+4a-1\right){x}{y}+\left(-a^{3}+2a^{2}+4a\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a\right){x}^{2}+\left(5a^{3}-8a^{2}-18a-6\right){x}+9a^{3}-18a^{2}-22a-4$ |
| 343.4-a4 |
343.4-a |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( - 7^{16} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$41.53148583$ |
3.202318121 |
\( -\frac{30631833226798148}{282475249} a^{3} + \frac{19913145044553620}{282475249} a^{2} + \frac{160003945294303955}{282475249} a + \frac{86891270576754902}{282475249} \) |
\( \bigl[-a^{3} + 2 a^{2} + 3 a\) , \( a\) , \( a^{3} - a^{2} - 4 a - 1\) , \( -7 a^{3} + 2 a^{2} + 14 a - 8\) , \( 69 a^{3} - 98 a^{2} - 255 a + 92\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+3a\right){x}{y}+\left(a^{3}-a^{2}-4a-1\right){y}={x}^{3}+a{x}^{2}+\left(-7a^{3}+2a^{2}+14a-8\right){x}+69a^{3}-98a^{2}-255a+92$ |
| 343.4-b1 |
343.4-b |
$1$ |
$1$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( 7^{14} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$21.23444099$ |
1.309838718 |
\( -\frac{349611581}{117649} a^{3} + \frac{303195395}{117649} a^{2} + \frac{1011828079}{117649} a - \frac{369510387}{117649} \) |
\( \bigl[a^{2} - a - 2\) , \( a^{2} - 3 a - 3\) , \( a^{2} - 2 a - 2\) , \( -11 a^{3} + 19 a^{2} + 39 a - 16\) , \( -23 a^{3} + 40 a^{2} + 84 a - 36\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-2a-2\right){y}={x}^{3}+\left(a^{2}-3a-3\right){x}^{2}+\left(-11a^{3}+19a^{2}+39a-16\right){x}-23a^{3}+40a^{2}+84a-36$ |
| 343.4-c1 |
343.4-c |
$1$ |
$1$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( 7^{14} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.263942447$ |
$47.27180061$ |
3.078565280 |
\( \frac{782646012}{117649} a^{3} - \frac{736229826}{117649} a^{2} - \frac{3610034665}{117649} a - \frac{1733848910}{117649} \) |
\( \bigl[a^{3} - a^{2} - 4 a\) , \( a^{3} - 2 a^{2} - 4 a + 1\) , \( a + 1\) , \( 63 a^{3} - 104 a^{2} - 249 a + 96\) , \( -332 a^{3} + 547 a^{2} + 1305 a - 515\bigr] \) |
${y}^2+\left(a^{3}-a^{2}-4a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}-2a^{2}-4a+1\right){x}^{2}+\left(63a^{3}-104a^{2}-249a+96\right){x}-332a^{3}+547a^{2}+1305a-515$ |
| 343.4-d1 |
343.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( 7^{26} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$13.08439948$ |
2.017765978 |
\( \frac{91176666325}{282475249} a^{3} - \frac{91176666325}{282475249} a^{2} - \frac{547059997950}{282475249} a + \frac{199910878122}{282475249} \) |
\( \bigl[a^{2} - a - 1\) , \( 2 a^{3} - 3 a^{2} - 9 a\) , \( a^{3} - a^{2} - 5 a\) , \( 3 a^{3} - 7 a^{2} - 7 a + 4\) , \( 44 a^{3} - 82 a^{2} - 339 a - 173\bigr] \) |
${y}^2+\left(a^{2}-a-1\right){x}{y}+\left(a^{3}-a^{2}-5a\right){y}={x}^{3}+\left(2a^{3}-3a^{2}-9a\right){x}^{2}+\left(3a^{3}-7a^{2}-7a+4\right){x}+44a^{3}-82a^{2}-339a-173$ |
| 343.4-d2 |
343.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( 7^{16} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$52.33759793$ |
2.017765978 |
\( -\frac{173650213}{16807} a^{3} + \frac{173650213}{16807} a^{2} + \frac{1041901278}{16807} a + \frac{583264453}{16807} \) |
\( \bigl[-a^{3} + 2 a^{2} + 4 a\) , \( a^{3} - a^{2} - 6 a - 1\) , \( -a^{3} + 2 a^{2} + 4 a\) , \( a^{3} + 4 a^{2} - 21 a - 5\) , \( 5 a^{2} - 15 a - 3\bigr] \) |
${y}^2+\left(-a^{3}+2a^{2}+4a\right){x}{y}+\left(-a^{3}+2a^{2}+4a\right){y}={x}^{3}+\left(a^{3}-a^{2}-6a-1\right){x}^{2}+\left(a^{3}+4a^{2}-21a-5\right){x}+5a^{2}-15a-3$ |
| 343.4-d3 |
343.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( 7^{8} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$1$ |
$261.6879896$ |
2.017765978 |
\( -\frac{94831363}{7} a^{3} + \frac{94831363}{7} a^{2} + \frac{568988178}{7} a + \frac{268859728}{7} \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( 90 a^{3} - 60 a^{2} - 466 a - 253\) , \( -608 a^{3} + 399 a^{2} + 3168 a + 1717\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-a^{3}+2a^{2}+3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(90a^{3}-60a^{2}-466a-253\right){x}-608a^{3}+399a^{2}+3168a+1717$ |
| 343.4-d4 |
343.4-d |
$4$ |
$10$ |
4.4.4205.1 |
$4$ |
$[4, 0]$ |
343.4 |
\( 7^{3} \) |
\( 7^{10} \) |
$12.02080$ |
$(-a^3+2a^2+3a-3), (a^2-2a-3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2^{3} \) |
$1$ |
$65.42199742$ |
2.017765978 |
\( \frac{213433415640625}{49} a^{3} - \frac{213433415640625}{49} a^{2} - \frac{1280600493843750}{49} a + \frac{467970351097797}{49} \) |
\( \bigl[a + 1\) , \( -a^{3} + a^{2} + 6 a + 1\) , \( -a^{3} + 2 a^{2} + 3 a - 1\) , \( 95 a^{3} - 90 a^{2} - 421 a - 253\) , \( -810 a^{3} + 800 a^{2} + 3657 a + 1724\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(-a^{3}+2a^{2}+3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+6a+1\right){x}^{2}+\left(95a^{3}-90a^{2}-421a-253\right){x}-810a^{3}+800a^{2}+3657a+1724$ |
*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.
