| 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 |
| 441.1-a1 |
441.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{12} \cdot 7^{4} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.014980985$ |
1.348340487 |
\( \frac{300763}{35721} \) |
\( \bigl[\phi\) , \( \phi - 1\) , \( \phi\) , \( -2 \phi + 3\) , \( 17 \phi - 30\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-2\phi+3\right){x}+17\phi-30$ |
| 441.1-a2 |
441.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{6} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$12.05992394$ |
1.348340487 |
\( \frac{5177717}{189} \) |
\( \bigl[\phi + 1\) , \( \phi\) , \( \phi\) , \( -3 \phi - 3\) , \( -9 \phi - 6\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\phi{y}={x}^{3}+\phi{x}^{2}+\left(-3\phi-3\right){x}-9\phi-6$ |
| 441.1-b1 |
441.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.04890414$ |
1.235305037 |
\( \frac{2967217}{189} a - \frac{14532341}{567} \) |
\( \bigl[\phi\) , \( -\phi\) , \( \phi\) , \( -3\) , \( -2 \phi\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}-\phi{x}^{2}-3{x}-2\phi$ |
| 441.1-b2 |
441.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$11.04890414$ |
1.235305037 |
\( -\frac{5300015616722532145}{7} a + \frac{25726816226286915413}{21} \) |
\( \bigl[\phi + 1\) , \( \phi - 1\) , \( \phi + 1\) , \( -142 \phi - 199\) , \( -670 \phi + 101\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}+\left(\phi-1\right){x}^{2}+\left(-142\phi-199\right){x}-670\phi+101$ |
| 441.1-b3 |
441.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{4} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$11.04890414$ |
1.235305037 |
\( -\frac{12214665265}{21} a + \frac{415283030098}{441} \) |
\( \bigl[\phi\) , \( -\phi\) , \( \phi\) , \( -48\) , \( -101 \phi + 54\bigr] \) |
${y}^2+\phi{x}{y}+\phi{y}={x}^{3}-\phi{x}^{2}-48{x}-101\phi+54$ |
| 441.1-b4 |
441.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{8} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.762226036$ |
1.235305037 |
\( \frac{46991733203041}{343} a + \frac{609892519727245}{7203} \) |
\( \bigl[1\) , \( -\phi + 1\) , \( \phi\) , \( 98 \phi - 269\) , \( -1112 \phi + 1370\bigr] \) |
${y}^2+{x}{y}+\phi{y}={x}^{3}+\left(-\phi+1\right){x}^{2}+\left(98\phi-269\right){x}-1112\phi+1370$ |
| 441.1-c1 |
441.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$11.04890414$ |
1.235305037 |
\( -\frac{2967217}{189} a - \frac{5630690}{567} \) |
\( \bigl[\phi + 1\) , \( -1\) , \( \phi + 1\) , \( -2 \phi - 3\) , \( \phi - 2\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-2\phi-3\right){x}+\phi-2$ |
| 441.1-c2 |
441.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{8} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$2.762226036$ |
1.235305037 |
\( -\frac{46991733203041}{343} a + \frac{1596718916991106}{7203} \) |
\( \bigl[1\) , \( \phi\) , \( \phi + 1\) , \( -99 \phi - 171\) , \( 1111 \phi + 258\bigr] \) |
${y}^2+{x}{y}+\left(\phi+1\right){y}={x}^{3}+\phi{x}^{2}+\left(-99\phi-171\right){x}+1111\phi+258$ |
| 441.1-c3 |
441.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{4} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$11.04890414$ |
1.235305037 |
\( \frac{12214665265}{21} a + \frac{158775059533}{441} \) |
\( \bigl[\phi + 1\) , \( -1\) , \( \phi + 1\) , \( -2 \phi - 48\) , \( 100 \phi - 47\bigr] \) |
${y}^2+\left(\phi+1\right){x}{y}+\left(\phi+1\right){y}={x}^{3}-{x}^{2}+\left(-2\phi-48\right){x}+100\phi-47$ |
| 441.1-c4 |
441.1-c |
$4$ |
$4$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$11.04890414$ |
1.235305037 |
\( \frac{5300015616722532145}{7} a + \frac{9826769376119318978}{21} \) |
\( \bigl[\phi\) , \( \phi + 1\) , \( 1\) , \( 142 \phi - 339\) , \( 471 \phi - 427\bigr] \) |
${y}^2+\phi{x}{y}+{y}={x}^{3}+\left(\phi+1\right){x}^{2}+\left(142\phi-339\right){x}+471\phi-427$ |
| 441.1-d1 |
441.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{16} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.814020435$ |
0.728082011 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \) |
${y}^2+{x}{y}={x}^{3}-34{x}-217$ |
| 441.1-d2 |
441.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$13.02432697$ |
0.728082011 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}$ |
| 441.1-d3 |
441.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{4} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$13.02432697$ |
0.728082011 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-4{x}-1$ |
| 441.1-d4 |
441.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{2} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$13.02432697$ |
0.728082011 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \) |
${y}^2+{x}{y}={x}^{3}-39{x}+90$ |
| 441.1-d5 |
441.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{8} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.256081743$ |
0.728082011 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \) |
${y}^2+{x}{y}={x}^{3}-49{x}-136$ |
| 441.1-d6 |
441.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{5}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{4} \) |
$0.91566$ |
$(3), (7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$1$ |
$0.814020435$ |
0.728082011 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \) |
${y}^2+{x}{y}={x}^{3}-784{x}-8515$ |
*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.
