| 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 |
| 146692.2-a1 |
146692.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{20} \cdot 7^{3} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.257585578$ |
$0.551804301$ |
3.282509488 |
\( \frac{10621452329}{10888192} a - \frac{7274546105}{10888192} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 138 a - 44\) , \( 685 a + 155\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(138a-44\right){x}+685a+155$ |
| 146692.2-a2 |
146692.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{10} \cdot 7^{6} \cdot 13^{6} \cdot 31^{2} \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 5 \) |
$0.515171157$ |
$0.275902150$ |
3.282509488 |
\( -\frac{6512659898044201}{3617942048} a + \frac{303261539125773}{452242756} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 2538 a - 1164\) , \( 28909 a + 16187\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(2538a-1164\right){x}+28909a+16187$ |
| 146692.2-b1 |
146692.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{6} \cdot 7^{6} \cdot 13^{6} \cdot 31^{3} \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs[2] |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.084213761$ |
$0.413568032$ |
2.895555473 |
\( -\frac{27687863199645}{14019525436} a - \frac{39320031191761}{28039050872} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 54 a + 271\) , \( -2763 a + 1641\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(54a+271\right){x}-2763a+1641$ |
| 146692.2-b2 |
146692.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 7^{18} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.252641283$ |
$0.137856010$ |
2.895555473 |
\( \frac{20687422086138241443}{50480821535223919} a + \frac{113695097195902805459}{100961643070447838} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -206 a - 2199\) , \( 27717 a - 3795\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-206a-2199\right){x}+27717a-3795$ |
| 146692.2-b3 |
146692.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 7^{2} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \) |
$0.252641283$ |
$1.240704096$ |
2.895555473 |
\( -\frac{1566729405}{3038} a + \frac{66816311}{3038} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 104 a - 34\) , \( -183 a - 239\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(104a-34\right){x}-183a-239$ |
| 146692.2-b4 |
146692.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 7^{2} \cdot 13^{6} \cdot 31^{9} \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$0.252641283$ |
$0.137856010$ |
2.895555473 |
\( \frac{406635051366709052855}{2591082971745758} a + \frac{44028606239712586643}{1295541485872879} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( -7296 a + 3701\) , \( -154103 a + 217381\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-7296a+3701\right){x}-154103a+217381$ |
| 146692.2-b5 |
146692.2-b |
$5$ |
$9$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{18} \cdot 7^{2} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B[2] |
$9$ |
\( 2^{2} \) |
$0.252641283$ |
$0.137856010$ |
2.895555473 |
\( -\frac{19926242340409933}{388864} a + \frac{18769373204677155}{777728} \) |
\( \bigl[1\) , \( a - 1\) , \( 1\) , \( 3614 a + 24016\) , \( -1825903 a + 1200467\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(3614a+24016\right){x}-1825903a+1200467$ |
| 146692.2-c1 |
146692.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{4} \cdot 7^{3} \cdot 13^{8} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.422296567$ |
$0.928062016$ |
5.430566850 |
\( -\frac{295041609}{7187908} a + \frac{107809434}{1796977} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -21 a + 13\) , \( -188 a + 73\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-21a+13\right){x}-188a+73$ |
| 146692.2-c2 |
146692.2-c |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{2} \cdot 7^{6} \cdot 13^{7} \cdot 31^{2} \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.844593135$ |
$0.464031008$ |
5.430566850 |
\( -\frac{233519701564899}{2939577914} a + \frac{172304309837943}{2939577914} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -191 a + 543\) , \( -3798 a - 45\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-191a+543\right){x}-3798a-45$ |
| 146692.2-d1 |
146692.2-d |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{22} \cdot 7^{4} \cdot 13^{8} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$1$ |
$0.239928512$ |
2.216364664 |
\( \frac{354220679986287}{25761462272} a - \frac{497011551977149}{25761462272} \) |
\( \bigl[a + 1\) , \( -a\) , \( a\) , \( -136 a - 1281\) , \( 3822 a + 17758\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-136a-1281\right){x}+3822a+17758$ |
| 146692.2-e1 |
146692.2-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
146692.2 |
\( 2^{2} \cdot 7 \cdot 13^{2} \cdot 31 \) |
\( 2^{10} \cdot 7^{2} \cdot 13^{6} \cdot 31 \) |
$3.02901$ |
$(-3a+1), (-4a+1), (6a-5), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$1.174084672$ |
5.422864813 |
\( \frac{8685387}{48608} a + \frac{17171919}{48608} \) |
\( \bigl[a\) , \( -1\) , \( 1\) , \( 21 a - 23\) , \( -19 a + 81\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(21a-23\right){x}-19a+81$ |
*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.
