| 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 |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{2} \cdot 7^{7} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$3.732136758$ |
1.386080795 |
\( -\frac{979400228864}{352947} a - \frac{709250383872}{117649} \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -17 a - 42\) , \( -57 a - 124\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-17a-42\right){x}-57a-124$ |
| 441.1-b1 |
441.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{10} \cdot 7^{3} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.306738992$ |
$14.52886078$ |
3.310255692 |
\( \frac{4947968}{11907} a + \frac{753664}{1701} \) |
\( \bigl[0\) , \( -a\) , \( a\) , \( 5 a - 13\) , \( -17 a + 51\bigr] \) |
${y}^2+a{y}={x}^{3}-a{x}^{2}+\left(5a-13\right){x}-17a+51$ |
| 441.1-c1 |
441.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{16} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$4.513857150$ |
$0.814020435$ |
1.364627489 |
\( -\frac{4354703137}{17294403} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -34\) , \( -217\bigr] \) |
${y}^2+{x}{y}={x}^{3}-34{x}-217$ |
| 441.1-c2 |
441.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{2} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.256928575$ |
$13.02432697$ |
1.364627489 |
\( \frac{103823}{63} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}$ |
| 441.1-c3 |
441.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{8} \cdot 7^{4} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.128464287$ |
$13.02432697$ |
1.364627489 |
\( \frac{7189057}{3969} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -4\) , \( -1\bigr] \) |
${y}^2+{x}{y}={x}^{3}-4{x}-1$ |
| 441.1-c4 |
441.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{16} \cdot 7^{2} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$2.256928575$ |
$13.02432697$ |
1.364627489 |
\( \frac{6570725617}{45927} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -39\) , \( 90\bigr] \) |
${y}^2+{x}{y}={x}^{3}-39{x}+90$ |
| 441.1-c5 |
441.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{4} \cdot 7^{8} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.256928575$ |
$3.256081743$ |
1.364627489 |
\( \frac{13027640977}{21609} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -49\) , \( -136\bigr] \) |
${y}^2+{x}{y}={x}^{3}-49{x}-136$ |
| 441.1-c6 |
441.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{2} \cdot 7^{4} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$4.513857150$ |
$0.814020435$ |
1.364627489 |
\( \frac{53297461115137}{147} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -784\) , \( -8515\bigr] \) |
${y}^2+{x}{y}={x}^{3}-784{x}-8515$ |
| 441.1-d1 |
441.1-d |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{10} \cdot 7^{3} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.306738992$ |
$14.52886078$ |
3.310255692 |
\( -\frac{4947968}{11907} a + \frac{3407872}{3969} \) |
\( \bigl[0\) , \( a - 1\) , \( a + 1\) , \( -5 a - 8\) , \( 16 a + 34\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a-8\right){x}+16a+34$ |
| 441.1-e1 |
441.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{2} \cdot 7^{7} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$3.732136758$ |
1.386080795 |
\( \frac{979400228864}{352947} a - \frac{443878768640}{50421} \) |
\( \bigl[0\) , \( a\) , \( a\) , \( 17 a - 59\) , \( 56 a - 180\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}+\left(17a-59\right){x}+56a-180$ |
| 441.1-f1 |
441.1-f |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{7} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.134404727$ |
$14.44452085$ |
4.326133632 |
\( -\frac{2527424512}{64827} a + \frac{401948672}{21609} \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -50 a - 111\) , \( 380 a + 833\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-50a-111\right){x}+380a+833$ |
| 441.1-g1 |
441.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( - 3^{6} \cdot 7^{7} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.134404727$ |
$14.44452085$ |
4.326133632 |
\( \frac{2527424512}{64827} a - \frac{188796928}{9261} \) |
\( \bigl[0\) , \( a - 1\) , \( a\) , \( 50 a - 161\) , \( -381 a + 1214\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(50a-161\right){x}-381a+1214$ |
| 441.1-h1 |
441.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{10} \cdot 7^{6} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.534773572$ |
2.353478179 |
\( -\frac{204290216}{583443} a + \frac{33794281}{83349} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -20 a + 58\) , \( 20 a - 67\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-20a+58\right){x}+20a-67$ |
| 441.1-h2 |
441.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{20} \cdot 7^{3} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.534773572$ |
2.353478179 |
\( \frac{311341610738}{964467} a + \frac{293591050901}{413343} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 85 a - 292\) , \( 545 a - 1768\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(85a-292\right){x}+545a-1768$ |
| 441.1-i1 |
441.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{10} \cdot 7^{6} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.534773572$ |
2.353478179 |
\( \frac{204290216}{583443} a + \frac{32269751}{583443} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 18 a + 38\) , \( -21 a - 47\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(18a+38\right){x}-21a-47$ |
| 441.1-i2 |
441.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{29}) \) |
$2$ |
$[2, 0]$ |
441.1 |
\( 3^{2} \cdot 7^{2} \) |
\( 3^{20} \cdot 7^{3} \) |
$2.20520$ |
$(-a), (a-1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.534773572$ |
2.353478179 |
\( -\frac{311341610738}{964467} a + \frac{2989162188521}{2893401} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -87 a - 207\) , \( -546 a - 1223\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-87a-207\right){x}-546a-1223$ |
*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.
