| 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 |
| 2205.1-a1 |
2205.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( - 3^{4} \cdot 5 \cdot 7^{12} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.765094738$ |
2.131015150 |
\( -\frac{1361254}{1715} a + \frac{6586475389}{5294205} \) |
\( \bigl[\phi\) , \( -\phi\) , \( 1\) , \( 37 \phi - 64\) , \( 9 \phi + 4\bigr] \) |
${y}^2+\phi{x}{y}+{y}={x}^{3}-\phi{x}^{2}+\left(37\phi-64\right){x}+9\phi+4$ |
| 2205.1-a2 |
2205.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{2} \cdot 7^{6} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.530189477$ |
2.131015150 |
\( -\frac{101689696}{5145} a + \frac{1144343597}{5145} \) |
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( 0\) , \( -20 \phi - 20\) , \( 52 \phi + 27\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-20\phi-20\right){x}+52\phi+27$ |
| 2205.1-b1 |
2205.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{4} \cdot 5^{8} \cdot 7^{4} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.056450715$ |
$4.896714470$ |
1.977922247 |
\( -\frac{22703491}{13125} a - \frac{23396906}{275625} \) |
\( \bigl[\phi + 1\) , \( \phi + 1\) , \( 1\) , \( -14 \phi + 1\) , \( 18 \phi + 2\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(-14\phi+1\right){x}+18\phi+2$ |
| 2205.1-b2 |
2205.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{4} \cdot 7^{2} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.112901431$ |
$19.58685788$ |
1.977922247 |
\( \frac{999695971}{175} a + \frac{371012669}{105} \) |
\( \bigl[\phi\) , \( 1\) , \( 1\) , \( 8 \phi - 17\) , \( 16 \phi - 25\bigr] \) |
${y}^2+\phi{x}{y}+{y}={x}^{3}+{x}^{2}+\left(8\phi-17\right){x}+16\phi-25$ |
| 2205.1-c1 |
2205.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( - 3^{4} \cdot 5 \cdot 7^{4} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.384977919$ |
$5.402041357$ |
1.860110393 |
\( -\frac{426314767}{105} a + \frac{14489372236}{2205} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( 0\) , \( 12 \phi + 4\) , \( -\phi - 7\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(12\phi+4\right){x}-\phi-7$ |
| 2205.1-c2 |
2205.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.192488959$ |
$21.60816543$ |
1.860110393 |
\( -\frac{2773}{3} a + \frac{341641}{105} \) |
\( \bigl[\phi + 1\) , \( 0\) , \( \phi\) , \( -2\) , \( -1\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}-2{x}-1$ |
| 2205.1-d1 |
2205.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.192488959$ |
$21.60816543$ |
1.860110393 |
\( \frac{2773}{3} a + \frac{244586}{105} \) |
\( \bigl[\phi\) , \( -\phi + 1\) , \( \phi + 1\) , \( -2 \phi - 1\) , \( -\phi - 1\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-2\phi-1\right){x}-\phi-1$ |
| 2205.1-d2 |
2205.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( - 3^{4} \cdot 5 \cdot 7^{4} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.384977919$ |
$5.402041357$ |
1.860110393 |
\( \frac{426314767}{105} a + \frac{5536762129}{2205} \) |
\( \bigl[1\) , \( \phi\) , \( 0\) , \( -12 \phi + 16\) , \( \phi - 8\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\phi{x}^{2}+\left(-12\phi+16\right){x}+\phi-8$ |
| 2205.1-e1 |
2205.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{8} \cdot 5^{2} \cdot 7^{8} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.457805429$ |
$2.148755400$ |
1.759717321 |
\( \frac{590589719}{972405} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 17\) , \( -37\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+17{x}-37$ |
| 2205.1-e2 |
2205.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{4} \cdot 5^{4} \cdot 7^{4} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$0.228902714$ |
$8.595021600$ |
1.759717321 |
\( \frac{47045881}{11025} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -8\) , \( -7\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-8{x}-7$ |
| 2205.1-e3 |
2205.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{2} \cdot 7^{2} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.457805429$ |
$34.38008640$ |
1.759717321 |
\( \frac{1771561}{105} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -3\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-3{x}+1$ |
| 2205.1-e4 |
2205.1-e |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{8} \cdot 7^{2} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.457805429$ |
$2.148755400$ |
1.759717321 |
\( \frac{157551496201}{13125} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -113\) , \( -469\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-113{x}-469$ |
| 2205.1-f1 |
2205.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( - 3^{4} \cdot 5 \cdot 7^{12} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.765094738$ |
2.131015150 |
\( \frac{1361254}{1715} a + \frac{2384284291}{5294205} \) |
\( \bigl[\phi + 1\) , \( -1\) , \( 1\) , \( -38 \phi - 27\) , \( -9 \phi + 13\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-38\phi-27\right){x}-9\phi+13$ |
| 2205.1-f2 |
2205.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{2} \cdot 7^{6} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$9.530189477$ |
2.131015150 |
\( \frac{101689696}{5145} a + \frac{1042653901}{5145} \) |
\( \bigl[\phi\) , \( \phi + 1\) , \( \phi + 1\) , \( 21 \phi - 41\) , \( -73 \phi + 99\bigr] \) |
${y}^2+\phi{x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(21\phi-41\right){x}-73\phi+99$ |
| 2205.1-g1 |
2205.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{4} \cdot 5^{8} \cdot 7^{4} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.056450715$ |
$4.896714470$ |
1.977922247 |
\( \frac{22703491}{13125} a - \frac{500170217}{275625} \) |
\( \bigl[\phi\) , \( \phi\) , \( 0\) , \( 17 \phi - 16\) , \( -33 \phi + 51\bigr] \) |
${y}^2+\phi{x}{y}={x}^{3}+\phi{x}^{2}+\left(17\phi-16\right){x}-33\phi+51$ |
| 2205.1-g2 |
2205.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
2205.1 |
\( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( 3^{2} \cdot 5^{4} \cdot 7^{2} \) |
$1.36923$ |
$(-2a+1), (3), (7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.112901431$ |
$19.58685788$ |
1.977922247 |
\( -\frac{999695971}{175} a + \frac{4854151258}{525} \) |
\( \bigl[\phi + 1\) , \( -\phi + 1\) , \( 1\) , \( -9 \phi - 9\) , \( -16 \phi - 9\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(-9\phi-9\right){x}-16\phi-9$ |
*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.
