| 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 |
| 34.1-a1 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{18} \cdot 17^{3} \) |
$0.61031$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.790055734$ |
0.493216832 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -16 a + 19\) , \( -25 a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-16a+19\right){x}-25a+31$ |
| 272.1-a1 |
272.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
272.1 |
\( 2^{4} \cdot 17 \) |
\( 2^{30} \cdot 17^{3} \) |
$1.02642$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.571768199$ |
1.262810757 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -61 a + 74\) , \( 276 a - 375\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-61a+74\right){x}+276a-375$ |
| 578.2-d1 |
578.2-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
578.2 |
\( 2 \cdot 17^{2} \) |
\( 2^{18} \cdot 17^{9} \) |
$1.23927$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$1.732562065$ |
0.612553192 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( 0\) , \( -201 a - 210\) , \( 1560 a + 2026\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-201a-210\right){x}+1560a+2026$ |
| 1666.3-a1 |
1666.3-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1666.3 |
\( 2 \cdot 7^{2} \cdot 17 \) |
\( 2^{18} \cdot 7^{6} \cdot 17^{3} \) |
$1.61474$ |
$(a), (-2a+1), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1$ |
$0.849008023$ |
2.701526987 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( -130 a - 166\) , \( -1023 a - 1469\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-130a-166\right){x}-1023a-1469$ |
| 1666.5-l1 |
1666.5-l |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
1666.5 |
\( 2 \cdot 7^{2} \cdot 17 \) |
\( 2^{18} \cdot 7^{6} \cdot 17^{3} \) |
$1.61474$ |
$(a), (2a+1), (-3a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3^{3} \) |
$0.026095048$ |
$3.353638960$ |
3.341590108 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -64 a + 45\) , \( -191 a + 409\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-64a+45\right){x}-191a+409$ |
| 2754.1-b1 |
2754.1-b |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
2754.1 |
\( 2 \cdot 3^{4} \cdot 17 \) |
\( 2^{18} \cdot 3^{12} \cdot 17^{3} \) |
$1.83094$ |
$(a), (-3a-1), (3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{3} \) |
$1$ |
$2.381178799$ |
2.525621514 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -137 a + 167\) , \( 1015 a - 1403\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-137a+167\right){x}+1015a-1403$ |
| 4352.1-g1 |
4352.1-g |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4352.1 |
\( 2^{8} \cdot 17 \) |
\( 2^{42} \cdot 17^{3} \) |
$2.05284$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.218223669$ |
2.352781498 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -140 a - 88\) , \( 456 a + 1080\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-140a-88\right){x}+456a+1080$ |
| 4352.1-k1 |
4352.1-k |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4352.1 |
\( 2^{8} \cdot 17 \) |
\( 2^{42} \cdot 17^{3} \) |
$2.05284$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$0.561566022$ |
0.595630713 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -140 a - 88\) , \( -456 a - 1080\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-140a-88\right){x}-456a-1080$ |
| 4624.3-d1 |
4624.3-d |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
4624.3 |
\( 2^{4} \cdot 17^{2} \) |
\( 2^{30} \cdot 17^{9} \) |
$2.08419$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B |
$9$ |
\( 2^{3} \) |
$1$ |
$0.338343955$ |
2.153207748 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -802 a - 845\) , \( -13283 a - 17051\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-802a-845\right){x}-13283a-17051$ |
*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.
