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 |
46.3-a1 |
46.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2 \cdot 23 \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$389.5837663$ |
0.901814273 |
\( -\frac{1966014850099606425355075}{46} a^{3} + \frac{1017684172320311147843853}{46} a^{2} + \frac{7337267309006641235101365}{46} a - \frac{3798049037158082012321527}{46} \) |
\( \bigl[a^{2} + a - 1\) , \( -a^{2} + a + 1\) , \( a^{3} - 3 a\) , \( 14 a^{3} + 28 a^{2} - 103 a - 60\) , \( -146 a^{3} + 37 a^{2} + 419 a + 187\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+\left(-a^{2}+a+1\right){x}^{2}+\left(14a^{3}+28a^{2}-103a-60\right){x}-146a^{3}+37a^{2}+419a+187$ |
46.3-a2 |
46.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2^{3} \cdot 23^{3} \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 1 \) |
$1$ |
$389.5837663$ |
0.901814273 |
\( -\frac{363132080045}{24334} a^{3} + \frac{187956544995}{24334} a^{2} + \frac{677616522870}{12167} a - \frac{350737167701}{12167} \) |
\( \bigl[a\) , \( -a + 1\) , \( a^{2} + a - 1\) , \( -a^{3} + a\) , \( 9 a^{3} + 18 a^{2} - 2 a - 5\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a^{3}+a\right){x}+9a^{3}+18a^{2}-2a-5$ |
46.3-a3 |
46.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2^{9} \cdot 23^{9} \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$4.809676127$ |
0.901814273 |
\( \frac{3815600232237530491335}{14409221291704} a^{3} - \frac{7371063772893741178385}{14409221291704} a^{2} - \frac{1022542227359311031345}{14409221291704} a + \frac{1974989958426698326323}{14409221291704} \) |
\( \bigl[1\) , \( a^{3} - a^{2} - 4 a + 1\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 11 a^{3} + 6 a^{2} - 40 a - 23\) , \( 32 a^{3} - a^{2} - 138 a - 33\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(a^{3}-a^{2}-4a+1\right){x}^{2}+\left(11a^{3}+6a^{2}-40a-23\right){x}+32a^{3}-a^{2}-138a-33$ |
46.3-b1 |
46.3-b |
$2$ |
$11$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2^{11} \cdot 23 \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\Z/11\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.1.1 |
$1$ |
\( 11 \) |
$1$ |
$903.8888112$ |
1.711910627 |
\( -\frac{239761277}{184} a^{3} - \frac{222189605}{92} a^{2} + \frac{45413053}{92} a + \frac{105311111}{184} \) |
\( \bigl[a^{3} - 4 a\) , \( a^{3} + a^{2} - 4 a - 2\) , \( a + 1\) , \( -2 a^{3} + 7 a + 2\) , \( -13 a^{3} - 7 a^{2} + 48 a + 25\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{3}+a^{2}-4a-2\right){x}^{2}+\left(-2a^{3}+7a+2\right){x}-13a^{3}-7a^{2}+48a+25$ |
46.3-b2 |
46.3-b |
$2$ |
$11$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2 \cdot 23^{11} \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.1.2 |
$121$ |
\( 11 \) |
$1$ |
$0.061736822$ |
1.711910627 |
\( \frac{1188212612376161652506351375283224919}{1905619515827854} a^{3} - \frac{2295450498832145956657614793163523455}{1905619515827854} a^{2} - \frac{318380609923977761851224492368277287}{1905619515827854} a + \frac{615064107427012847029176371880971317}{1905619515827854} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( -1\) , \( a^{2} - 2\) , \( -662 a^{3} + 475 a^{2} + 2528 a - 1969\) , \( -17976 a^{3} + 10669 a^{2} + 67414 a - 41406\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}-{x}^{2}+\left(-662a^{3}+475a^{2}+2528a-1969\right){x}-17976a^{3}+10669a^{2}+67414a-41406$ |
46.3-c1 |
46.3-c |
$2$ |
$11$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2^{11} \cdot 23 \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.10.1 |
$1$ |
\( 1 \) |
$1$ |
$38.74391961$ |
0.807164991 |
\( -\frac{239761277}{184} a^{3} - \frac{222189605}{92} a^{2} + \frac{45413053}{92} a + \frac{105311111}{184} \) |
\( \bigl[a\) , \( -a^{3} - a^{2} + 4 a + 1\) , \( a^{3} + a^{2} - 4 a - 2\) , \( -2 a^{3} + 6 a + 2\) , \( 14 a^{3} + 7 a^{2} - 52 a - 27\bigr] \) |
${y}^2+a{x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(-a^{3}-a^{2}+4a+1\right){x}^{2}+\left(-2a^{3}+6a+2\right){x}+14a^{3}+7a^{2}-52a-27$ |
46.3-c2 |
46.3-c |
$2$ |
$11$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2 \cdot 23^{11} \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$11$ |
11B.10.2 |
$1$ |
\( 1 \) |
$1$ |
$38.74391961$ |
0.807164991 |
\( \frac{1188212612376161652506351375283224919}{1905619515827854} a^{3} - \frac{2295450498832145956657614793163523455}{1905619515827854} a^{2} - \frac{318380609923977761851224492368277287}{1905619515827854} a + \frac{615064107427012847029176371880971317}{1905619515827854} \) |
\( \bigl[a^{2} + a - 1\) , \( -a^{2} - a + 3\) , \( a^{2} + a - 2\) , \( -663 a^{3} + 474 a^{2} + 2530 a - 1966\) , \( 17313 a^{3} - 10194 a^{2} - 64883 a + 39437\bigr] \) |
${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{2}+a-2\right){y}={x}^{3}+\left(-a^{2}-a+3\right){x}^{2}+\left(-663a^{3}+474a^{2}+2530a-1966\right){x}+17313a^{3}-10194a^{2}-64883a+39437$ |
46.3-d1 |
46.3-d |
$3$ |
$9$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2 \cdot 23 \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$81$ |
\( 1 \) |
$1$ |
$1.131744228$ |
1.909818385 |
\( -\frac{1966014850099606425355075}{46} a^{3} + \frac{1017684172320311147843853}{46} a^{2} + \frac{7337267309006641235101365}{46} a - \frac{3798049037158082012321527}{46} \) |
\( \bigl[a^{3} + a^{2} - 4 a - 1\) , \( a + 1\) , \( a^{3} + a^{2} - 4 a - 2\) , \( 15 a^{3} + 27 a^{2} - 104 a - 61\) , \( 188 a^{3} - 56 a^{2} - 584 a - 263\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-4a-1\right){x}{y}+\left(a^{3}+a^{2}-4a-2\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15a^{3}+27a^{2}-104a-61\right){x}+188a^{3}-56a^{2}-584a-263$ |
46.3-d2 |
46.3-d |
$3$ |
$9$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2^{3} \cdot 23^{3} \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 3^{2} \) |
$1$ |
$91.67128249$ |
1.909818385 |
\( -\frac{363132080045}{24334} a^{3} + \frac{187956544995}{24334} a^{2} + \frac{677616522870}{12167} a - \frac{350737167701}{12167} \) |
\( \bigl[a^{3} - 4 a\) , \( a + 1\) , \( a^{2} - 1\) , \( a + 1\) , \( -10 a^{3} - 19 a^{2} + 4 a + 5\bigr] \) |
${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+1\right){x}-10a^{3}-19a^{2}+4a+5$ |
46.3-d3 |
46.3-d |
$3$ |
$9$ |
\(\Q(\sqrt{2}, \sqrt{3})\) |
$4$ |
$[4, 0]$ |
46.3 |
\( 2 \cdot 23 \) |
\( 2^{9} \cdot 23^{9} \) |
$6.92189$ |
$(a^3-4a+1), (-a^2+a+3)$ |
0 |
$\Z/9\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 3^{4} \) |
$1$ |
$91.67128249$ |
1.909818385 |
\( \frac{3815600232237530491335}{14409221291704} a^{3} - \frac{7371063772893741178385}{14409221291704} a^{2} - \frac{1022542227359311031345}{14409221291704} a + \frac{1974989958426698326323}{14409221291704} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} + a^{2} + 4 a - 2\) , \( a^{3} + a^{2} - 3 a - 1\) , \( 12 a^{3} + 6 a^{2} - 44 a - 24\) , \( -32 a^{3} + 136 a + 32\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}+a^{2}-3a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-2\right){x}^{2}+\left(12a^{3}+6a^{2}-44a-24\right){x}-32a^{3}+136a+32$ |
*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.