| 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 |
| 9.1-a1 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{20} \) |
$0.43777$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.266923342$ |
0.223962521 |
\( -\frac{873722816}{59049} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -40 a - 60\) , \( -153 a - 220\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-40a-60\right){x}-153a-220$ |
| 9.1-a2 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{4} \) |
$0.43777$ |
$(3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$31.67308356$ |
0.223962521 |
\( \frac{64}{9} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}$ |
| 9.1-a3 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{2} \) |
$0.43777$ |
$(3)$ |
0 |
$\Z/10\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$63.34616712$ |
0.223962521 |
\( \frac{85184}{3} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -2 a - 3\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-2a-3\right){x}$ |
| 9.1-a4 |
9.1-a |
$4$ |
$10$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
9.1 |
\( 3^{2} \) |
\( 3^{10} \) |
$0.43777$ |
$(3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.533846685$ |
0.223962521 |
\( \frac{58591911104}{243} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -162 a - 243\) , \( -1495 a - 2130\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-162a-243\right){x}-1495a-2130$ |
| 17.1-a1 |
17.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{2} \) |
$0.51321$ |
$(-3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$22.23161552$ |
0.436670169 |
\( -\frac{94464}{289} a + \frac{58688}{289} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -2 a + 1\) , \( -a\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a+1\right){x}-a$ |
| 17.1-a2 |
17.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{6} \) |
$0.51321$ |
$(-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.470179502$ |
0.436670169 |
\( \frac{1522678220544}{24137569} a - \frac{2147745195712}{24137569} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( 13 a - 19\) , \( 47 a - 69\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(13a-19\right){x}+47a-69$ |
| 17.1-a3 |
17.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17^{3} \) |
$0.51321$ |
$(-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.940359004$ |
0.436670169 |
\( -\frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -34 a - 53\) , \( -113 a - 163\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-34a-53\right){x}-113a-163$ |
| 17.1-a4 |
17.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.1 |
\( 17 \) |
\( 17 \) |
$0.51321$ |
$(-3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$44.46323104$ |
0.436670169 |
\( \frac{3690752}{17} a + \frac{5287232}{17} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( a - 1\) , \( -1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a-1\right){x}-1$ |
| 17.2-a1 |
17.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{6} \) |
$0.51321$ |
$(3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.470179502$ |
0.436670169 |
\( -\frac{1522678220544}{24137569} a - \frac{2147745195712}{24137569} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( -14 a - 19\) , \( -48 a - 69\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-14a-19\right){x}-48a-69$ |
| 17.2-a2 |
17.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{2} \) |
$0.51321$ |
$(3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$22.23161552$ |
0.436670169 |
\( \frac{94464}{289} a + \frac{58688}{289} \) |
\( \bigl[a\) , \( a + 1\) , \( a + 1\) , \( a + 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}$ |
| 17.2-a3 |
17.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17 \) |
$0.51321$ |
$(3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$44.46323104$ |
0.436670169 |
\( -\frac{3690752}{17} a + \frac{5287232}{17} \) |
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -2 a - 1\) , \( -a - 1\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-2a-1\right){x}-a-1$ |
| 17.2-a4 |
17.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
17.2 |
\( 17 \) |
\( 17^{3} \) |
$0.51321$ |
$(3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$4.940359004$ |
0.436670169 |
\( \frac{55615383938816}{4913} a + \frac{78652069441856}{4913} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( 33 a - 53\) , \( 112 a - 163\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(33a-53\right){x}+112a-163$ |
| 28.1-a1 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{12} \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( -\frac{29518306565684}{13841287201} a + \frac{41622722395132}{13841287201} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -9 a - 18\) , \( -320 a - 467\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-9a-18\right){x}-320a-467$ |
| 28.1-a2 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{4} \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/12\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( -\frac{861093316}{2401} a + \frac{1217791012}{2401} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( a + 2\) , \( 12 a + 17\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(a+2\right){x}+12a+17$ |
| 28.1-a3 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{3} \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( -\frac{91481168031853524}{343} a + \frac{129373908533024396}{343} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -19 a - 148\) , \( 318 a - 201\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-19a-148\right){x}+318a-201$ |
| 28.1-a4 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7 \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( \frac{4096}{7} a + \frac{16384}{7} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -2 a + 3\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-2a+3\right){x}$ |
| 28.1-a5 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$38.79793896$ |
0.571547619 |
\( \frac{435744}{49} a + \frac{712688}{49} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 2 a - 4\) , \( -a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2a-4\right){x}-a+1$ |
| 28.1-a6 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{6} \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$4.310882107$ |
0.571547619 |
\( -\frac{1137747277344}{117649} a + \frac{1622386617968}{117649} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -34 a - 53\) , \( -133 a - 203\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-34a-53\right){x}-133a-203$ |
| 28.1-a7 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7 \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( \frac{1720664028}{7} a + \frac{2434028852}{7} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 17 a - 29\) , \( 49 a - 66\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(17a-29\right){x}+49a-66$ |
| 28.1-a8 |
28.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{3} \) |
$0.58140$ |
$(a), (-2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( \frac{1545435312128}{343} a + \frac{2185574023168}{343} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 18 a - 37\) , \( 68 a - 108\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(18a-37\right){x}+68a-108$ |
| 28.2-a1 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{3} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( -\frac{1545435312128}{343} a + \frac{2185574023168}{343} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -18 a - 37\) , \( -68 a - 108\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-18a-37\right){x}-68a-108$ |
| 28.2-a2 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7 \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( -\frac{4096}{7} a + \frac{16384}{7} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 2 a + 3\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(2a+3\right){x}$ |
| 28.2-a3 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{2} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$38.79793896$ |
0.571547619 |
\( -\frac{435744}{49} a + \frac{712688}{49} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -2 a - 4\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2a-4\right){x}+a+1$ |
| 28.2-a4 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{12} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( \frac{29518306565684}{13841287201} a + \frac{41622722395132}{13841287201} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 9 a - 18\) , \( 320 a - 467\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(9a-18\right){x}+320a-467$ |
| 28.2-a5 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7 \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( -\frac{1720664028}{7} a + \frac{2434028852}{7} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -17 a - 29\) , \( -49 a - 66\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-17a-29\right){x}-49a-66$ |
| 28.2-a6 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( - 2^{8} \cdot 7^{4} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/12\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$19.39896948$ |
0.571547619 |
\( \frac{861093316}{2401} a + \frac{1217791012}{2401} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -a + 2\) , \( -12 a + 17\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-a+2\right){x}-12a+17$ |
| 28.2-a7 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{6} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$4.310882107$ |
0.571547619 |
\( \frac{1137747277344}{117649} a + \frac{1622386617968}{117649} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 34 a - 53\) , \( 133 a - 203\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(34a-53\right){x}+133a-203$ |
| 28.2-a8 |
28.2-a |
$8$ |
$12$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{3} \) |
$0.58140$ |
$(a), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$2.155441053$ |
0.571547619 |
\( \frac{91481168031853524}{343} a + \frac{129373908533024396}{343} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 19 a - 148\) , \( -318 a - 201\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(19a-148\right){x}-318a-201$ |
| 31.1-a1 |
31.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( - 31^{4} \) |
$0.59638$ |
$(-4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.544454463$ |
0.449800251 |
\( -\frac{78588372777605}{923521} a + \frac{111138046783591}{923521} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -7 a - 21\) , \( -31 a - 52\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7a-21\right){x}-31a-52$ |
| 31.1-a2 |
31.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( -31 \) |
$0.59638$ |
$(-4a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$20.35563570$ |
0.449800251 |
\( \frac{47780}{31} a + \frac{69151}{31} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a + 1\) , \( -2 a - 1\) , \( -a - 2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-2a-1\right){x}-a-2$ |
| 31.1-a3 |
31.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( 31^{2} \) |
$0.59638$ |
$(-4a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$10.17781785$ |
0.449800251 |
\( \frac{4423034250}{961} a + \frac{6270751283}{961} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 10 a - 14\) , \( 21 a - 30\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(10a-14\right){x}+21a-30$ |
| 31.1-a4 |
31.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.1 |
\( 31 \) |
\( 31 \) |
$0.59638$ |
$(-4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$5.088908926$ |
0.449800251 |
\( \frac{1861812264958875}{31} a + \frac{2633000155833143}{31} \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( 25 a - 49\) , \( -79 a + 100\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(25a-49\right){x}-79a+100$ |
| 31.2-a1 |
31.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( -31 \) |
$0.59638$ |
$(4a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$20.35563570$ |
0.449800251 |
\( -\frac{47780}{31} a + \frac{69151}{31} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}$ |
| 31.2-a2 |
31.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( 31 \) |
$0.59638$ |
$(4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$5.088908926$ |
0.449800251 |
\( -\frac{1861812264958875}{31} a + \frac{2633000155833143}{31} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -25 a - 49\) , \( 79 a + 100\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-25a-49\right){x}+79a+100$ |
| 31.2-a3 |
31.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( 31^{2} \) |
$0.59638$ |
$(4a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$10.17781785$ |
0.449800251 |
\( -\frac{4423034250}{961} a + \frac{6270751283}{961} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -10 a - 14\) , \( -21 a - 30\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-10a-14\right){x}-21a-30$ |
| 31.2-a4 |
31.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
31.2 |
\( 31 \) |
\( - 31^{4} \) |
$0.59638$ |
$(4a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$2.544454463$ |
0.449800251 |
\( \frac{78588372777605}{923521} a + \frac{111138046783591}{923521} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( 5 a - 20\) , \( 10 a - 40\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(5a-20\right){x}+10a-40$ |
| 32.1-a1 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{9} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$3.437592909$ |
0.607686314 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( 15 a - 22\) , \( 46 a - 69\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(15a-22\right){x}+46a-69$ |
| 32.1-a2 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{9} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$27.50074327$ |
0.607686314 |
\( -29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 15 a - 23\) , \( -31 a + 46\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(15a-23\right){x}-31a+46$ |
| 32.1-a3 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$13.75037163$ |
0.607686314 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}$ |
| 32.1-a4 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{12} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-4$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$27.50074327$ |
0.607686314 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}$ |
| 32.1-a5 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$13.75037163$ |
0.607686314 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -2\) , \( -3\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-2{x}-3$ |
| 32.1-a6 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( 2^{6} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{potential}$ |
$-16$ |
$N(\mathrm{U}(1))$ |
✓ |
✓ |
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$55.00148654$ |
0.607686314 |
\( 287496 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -3\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}-3{x}$ |
| 32.1-a7 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{9} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$3.437592909$ |
0.607686314 |
\( 29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -15 a - 22\) , \( -46 a - 69\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-15a-22\right){x}-46a-69$ |
| 32.1-a8 |
32.1-a |
$8$ |
$16$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
32.1 |
\( 2^{5} \) |
\( - 2^{9} \) |
$0.60113$ |
$(a)$ |
0 |
$\Z/4\Z$ |
$\textsf{potential}$ |
$-64$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$27.50074327$ |
0.607686314 |
\( 29071392966 a + 41113158120 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -15 a - 23\) , \( 31 a + 46\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-15a-23\right){x}+31a+46$ |
| 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 |
\( -\frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -16 a + 19\) , \( -25 a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-16a+19\right){x}-25a+31$ |
| 34.1-a2 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 17 \) |
$0.61031$ |
$(a), (-3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$25.11050160$ |
0.493216832 |
\( \frac{2939047}{68} a - \frac{8089117}{136} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 2 a + 4\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a+4\right){x}$ |
| 34.1-a3 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{3} \cdot 17^{2} \) |
$0.61031$ |
$(a), (-3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$25.11050160$ |
0.493216832 |
\( -\frac{6018090689657}{1156} a + \frac{4255433785057}{578} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -8 a - 16\) , \( -10 a - 4\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-8a-16\right){x}-10a-4$ |
| 34.1-a4 |
34.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
34.1 |
\( 2 \cdot 17 \) |
\( 2^{9} \cdot 17^{6} \) |
$0.61031$ |
$(a), (-3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.790055734$ |
0.493216832 |
\( \frac{130087595511310753}{772402208} a + \frac{91987087285468997}{386201104} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 64 a - 141\) , \( -457 a + 479\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(64a-141\right){x}-457a+479$ |
| 34.2-a1 |
34.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
34.2 |
\( 2 \cdot 17 \) |
\( 2^{6} \cdot 17 \) |
$0.61031$ |
$(a), (3a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$25.11050160$ |
0.493216832 |
\( -\frac{2939047}{68} a - \frac{8089117}{136} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( -2 a + 4\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a+4\right){x}$ |
| 34.2-a2 |
34.2-a |
$4$ |
$6$ |
\(\Q(\sqrt{2}) \) |
$2$ |
$[2, 0]$ |
34.2 |
\( 2 \cdot 17 \) |
\( 2^{18} \cdot 17^{3} \) |
$0.61031$ |
$(a), (3a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$2.790055734$ |
0.493216832 |
\( \frac{9756993259}{1257728} a - \frac{25455932221}{2515456} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( 15 a + 20\) , \( 44 a + 62\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(15a+20\right){x}+44a+62$ |
*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.
