| 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 |
| 3751.6-a1 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( - 11^{6} \cdot 31 \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.787239209$ |
2.588132054 |
\( -\frac{106208}{31} a + \frac{51455}{31} \) |
\( \bigl[\phi\) , \( 1\) , \( 0\) , \( 2 \phi - 7\) , \( -5 \phi + 1\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+{x}^{2}+\left(2\phi-7\right){x}-5\phi+1$ |
| 3751.6-a2 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( - 11^{6} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.361702450$ |
2.588132054 |
\( -\frac{61725871986044215714}{961} a + \frac{99874558858644938523}{961} \) |
\( \bigl[\phi + 1\) , \( \phi + 1\) , \( 0\) , \( 1324 \phi - 1003\) , \( 2987 \phi - 28326\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(1324\phi-1003\right){x}+2987\phi-28326$ |
| 3751.6-a3 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{4} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$1.446809802$ |
2.588132054 |
\( -\frac{156520379364360}{923521} a + \frac{253260256463213}{923521} \) |
\( \bigl[\phi + 1\) , \( \phi + 1\) , \( 0\) , \( -201 \phi - 238\) , \( -2077 \phi - 1767\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(-201\phi-238\right){x}-2077\phi-1767$ |
| 3751.6-a4 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( - 11^{6} \cdot 31^{8} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.723404901$ |
2.588132054 |
\( \frac{11889611722383394}{852891037441} a - \frac{8629385062119691}{852891037441} \) |
\( \bigl[\phi\) , \( 1\) , \( 0\) , \( 167 \phi - 667\) , \( 5143 \phi - 5708\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+{x}^{2}+\left(167\phi-667\right){x}+5143\phi-5708$ |
| 3751.6-a5 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{6} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$4$ |
\( 2^{3} \) |
$1$ |
$2.893619604$ |
2.588132054 |
\( \frac{9029272560}{961} a + \frac{5599830233}{961} \) |
\( \bigl[\phi\) , \( 1\) , \( 0\) , \( -13 \phi - 57\) , \( -75 \phi - 192\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+{x}^{2}+\left(-13\phi-57\right){x}-75\phi-192$ |
| 3751.6-a6 |
3751.6-a |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( - 11^{6} \cdot 31 \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$0.723404901$ |
2.588132054 |
\( \frac{6130703730739448}{31} a + \frac{3788983280553597}{31} \) |
\( \bigl[1\) , \( -\phi - 1\) , \( 1\) , \( -173 \phi + 97\) , \( 3637 \phi - 6703\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-\phi-1\right){x}^{2}+\left(-173\phi+97\right){x}+3637\phi-6703$ |
| 3751.6-b1 |
3751.6-b |
$1$ |
$1$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{10} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$1$ |
$2.350210058$ |
2.102091781 |
\( \frac{80925680517}{961} a - \frac{130940503562}{961} \) |
\( \bigl[\phi\) , \( \phi\) , \( \phi + 1\) , \( -6 \phi - 93\) , \( -5426 \phi - 3012\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-6\phi-93\right){x}-5426\phi-3012$ |
| 3751.6-c1 |
3751.6-c |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{8} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$5.934044888$ |
1.769190366 |
\( -\frac{4849405}{961} a - \frac{2510893}{961} \) |
\( \bigl[\phi + 1\) , \( -1\) , \( 0\) , \( -20 \phi + 17\) , \( -28 \phi + 67\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}-{x}^{2}+\left(-20\phi+17\right){x}-28\phi+67$ |
| 3751.6-c2 |
3751.6-c |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{8} \cdot 31^{6} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.659338320$ |
1.769190366 |
\( \frac{939743984965}{887503681} a + \frac{11690363302}{887503681} \) |
\( \bigl[1\) , \( \phi\) , \( 0\) , \( 306 \phi + 149\) , \( 177 \phi - 74\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\phi{x}^{2}+\left(306\phi+149\right){x}+177\phi-74$ |
| 3751.6-d1 |
3751.6-d |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( - 11^{8} \cdot 31 \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$5.160184302$ |
2.307704575 |
\( -\frac{1427105}{3751} a + \frac{6573903}{3751} \) |
\( \bigl[1\) , \( -\phi\) , \( \phi\) , \( 11 \phi + 2\) , \( -14 \phi - 12\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-\phi{x}^{2}+\left(11\phi+2\right){x}-14\phi-12$ |
| 3751.6-d2 |
3751.6-d |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( - 11^{10} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.580092151$ |
2.307704575 |
\( \frac{6700954805}{14070001} a + \frac{25215080128}{14070001} \) |
\( \bigl[1\) , \( -\phi\) , \( \phi\) , \( -54 \phi - 13\) , \( -75 \phi - 54\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}-\phi{x}^{2}+\left(-54\phi-13\right){x}-75\phi-54$ |
| 3751.6-e1 |
3751.6-e |
$1$ |
$1$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{4} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2 \) |
$0.089879754$ |
$14.06152609$ |
2.260837373 |
\( \frac{80925680517}{961} a - \frac{130940503562}{961} \) |
\( \bigl[1\) , \( -\phi\) , \( 1\) , \( 2 \phi - 10\) , \( 88 \phi + 65\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-\phi{x}^{2}+\left(2\phi-10\right){x}+88\phi+65$ |
| 3751.6-f1 |
3751.6-f |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{2} \cdot 31^{2} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.196252751$ |
$6.612434719$ |
2.321411555 |
\( -\frac{4849405}{961} a - \frac{2510893}{961} \) |
\( \bigl[\phi\) , \( -\phi\) , \( \phi + 1\) , \( -3 \phi + 2\) , \( -3 \phi + 3\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}-\phi{x}^{2}+\left(-3\phi+2\right){x}-3\phi+3$ |
| 3751.6-f2 |
3751.6-f |
$2$ |
$3$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
3751.6 |
\( 11^{2} \cdot 31 \) |
\( 11^{2} \cdot 31^{6} \) |
$1.56372$ |
$(-3a+1), (5a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.588758255$ |
$2.204144906$ |
2.321411555 |
\( \frac{939743984965}{887503681} a + \frac{11690363302}{887503681} \) |
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( \phi\) , \( 21 \phi + 9\) , \( 19 \phi + 6\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(21\phi+9\right){x}+19\phi+6$ |
*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.
