| 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 |
| 574.2-a1 |
574.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( 2^{9} \cdot 7^{2} \cdot 41 \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \cdot 3^{2} \) |
$0.007812950$ |
$22.56836413$ |
2.244257402 |
\( -\frac{207187941}{64288} a - \frac{251550253}{32144} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 2 a - 2\) , \( -2 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(2a-2\right){x}-2a+3$ |
| 574.2-b1 |
574.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( 2^{3} \cdot 7^{6} \cdot 41 \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$0.464049747$ |
0.984398168 |
\( -\frac{779137480533106307}{19294436} a - \frac{550933703670956493}{9647218} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -89 a - 196\) , \( -682 a - 1234\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-89a-196\right){x}-682a-1234$ |
| 574.2-b2 |
574.2-b |
$2$ |
$3$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( 2 \cdot 7^{2} \cdot 41^{3} \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$4.176447723$ |
0.984398168 |
\( -\frac{13496525597}{6754258} a - \frac{4463300313}{3377129} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -4 a - 1\) , \( -2 a - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4a-1\right){x}-2a-2$ |
| 574.2-c1 |
574.2-c |
$1$ |
$1$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( 2^{13} \cdot 7 \cdot 41 \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.334032982$ |
1.178758665 |
\( \frac{278745991829}{36736} a - \frac{531794797629}{9184} \) |
\( \bigl[a + 1\) , \( -a\) , \( 1\) , \( -58 a - 123\) , \( 439 a + 496\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(-58a-123\right){x}+439a+496$ |
| 574.2-d1 |
574.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( - 2^{12} \cdot 7 \cdot 41 \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$18.13983194$ |
2.137799695 |
\( -\frac{239409963}{9184} a - \frac{620645219}{18368} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -a - 6\) , \( 2 a\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-6\right){x}+2a$ |
| 574.2-d2 |
574.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( - 2^{4} \cdot 7^{3} \cdot 41^{3} \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.015536882$ |
2.137799695 |
\( -\frac{187931084625206169}{47279806} a + \frac{531549483716434717}{94559612} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 119 a - 146\) , \( 774 a - 1052\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(119a-146\right){x}+774a-1052$ |
| 574.2-d3 |
574.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( - 2^{2} \cdot 7^{6} \cdot 41^{6} \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$1.007768441$ |
2.137799695 |
\( \frac{42628603748130403337}{1117690027698818} a - \frac{57394657600556843839}{1117690027698818} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( 69 a - 216\) , \( 842 a - 966\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(69a-216\right){x}+842a-966$ |
| 574.2-d4 |
574.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( - 2^{6} \cdot 7^{2} \cdot 41^{2} \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$9.069915970$ |
2.137799695 |
\( \frac{1187585012908957}{658952} a + \frac{1679499437585159}{658952} \) |
\( \bigl[a + 1\) , \( 0\) , \( a\) , \( -41 a - 46\) , \( 130 a + 168\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-41a-46\right){x}+130a+168$ |
| 574.2-e1 |
574.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
574.2 |
\( 2 \cdot 7 \cdot 41 \) |
\( 2^{3} \cdot 7^{5} \cdot 41 \) |
$1.23712$ |
$(a), (2a+1), (2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$3.097597433$ |
1.095166075 |
\( \frac{51683710549}{2756348} a - \frac{18098707713}{689087} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 10 a - 14\) , \( 21 a - 30\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(10a-14\right){x}+21a-30$ |
*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.
