| 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 |
| 20.1-a1 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{8} \cdot 5^{2} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$56.24286296$ |
0.770522476 |
\( \frac{13838352792668}{25} a^{2} - \frac{34335804621874}{25} a + \frac{9343051795654}{25} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - 1\) , \( 1292 a^{2} + 1512 a - 596\) , \( 38980968 a^{2} + 45611104 a - 17962864\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(1292a^{2}+1512a-596\right){x}+38980968a^{2}+45611104a-17962864$ |
| 20.1-a2 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{8} \cdot 5^{6} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$2.083068998$ |
0.770522476 |
\( -\frac{8271321267405818624506172}{15625} a^{2} + \frac{5698048560343235071057746}{15625} a + \frac{26586671253641020172342134}{15625} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - 1\) , \( -11628 a^{2} - 13608 a + 5354\) , \( -1052486900 a^{2} - 1231500702 a + 484997670\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-11628a^{2}-13608a+5354\right){x}-1052486900a^{2}-1231500702a+484997670$ |
| 20.1-a3 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$224.9714518$ |
0.770522476 |
\( \frac{185785252}{625} a^{2} - \frac{445080936}{625} a + \frac{123160356}{625} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - 1\) , \( -56443 a^{2} - 66043 a + 26009\) , \( 13218157 a^{2} + 15466387 a - 6091074\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-56443a^{2}-66043a+26009\right){x}+13218157a^{2}+15466387a-6091074$ |
| 20.1-a4 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{12} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$8.332275994$ |
0.770522476 |
\( -\frac{3077332109418143868}{244140625} a^{2} + \frac{2119953522630977224}{244140625} a + \frac{9891535601060896996}{244140625} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - 1\) , \( -516708 a^{2} - 604593 a + 238104\) , \( -366738953 a^{2} - 429116293 a + 168997390\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-516708a^{2}-604593a+238104\right){x}-366738953a^{2}-429116293a+168997390$ |
| 20.1-a5 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{8} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$56.24286296$ |
0.770522476 |
\( \frac{148445904580292}{390625} a^{2} + \frac{173694846086594}{390625} a - \frac{68405381268774}{390625} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - 1\) , \( -898838 a^{2} - 1051718 a + 414194\) , \( 854513176 a^{2} + 999854320 a - 393769180\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-898838a^{2}-1051718a+414194\right){x}+854513176a^{2}+999854320a-393769180$ |
| 20.1-a6 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{24} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$2.083068998$ |
0.770522476 |
\( \frac{3678043576600698623452}{59604644775390625} a^{2} - \frac{2506193844958703814786}{59604644775390625} a - \frac{11746914365786968746294}{59604644775390625} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a^{2} - 1\) , \( -1057168 a^{2} - 1236978 a + 487154\) , \( 532872492 a^{2} + 623506902 a - 245553574\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(-1057168a^{2}-1236978a+487154\right){x}+532872492a^{2}+623506902a-245553574$ |
| 20.1-a7 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{4} \cdot 5 \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$112.4857259$ |
0.770522476 |
\( -\frac{385146932}{5} a^{2} + \frac{268247176}{5} a + \frac{1242320604}{5} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a + 1\) , \( -101641 a^{2} - 118928 a + 46839\) , \( -32483626 a^{2} - 38008652 a + 14968816\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-101641a^{2}-118928a+46839\right){x}-32483626a^{2}-38008652a+14968816$ |
| 20.1-a8 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( - 2^{4} \cdot 5^{3} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$4.166137997$ |
0.770522476 |
\( \frac{179143376765057962508}{125} a^{2} + \frac{209613244321350839256}{125} a - \frac{82551261375569819076}{125} \) |
\( \bigl[a^{2} - a - 2\) , \( 0\) , \( a + 1\) , \( -8231946 a^{2} - 9632088 a + 3793374\) , \( -23684559407 a^{2} - 27712982904 a + 10914108517\bigr] \) |
${y}^2+\left(a^{2}-a-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-8231946a^{2}-9632088a+3793374\right){x}-23684559407a^{2}-27712982904a+10914108517$ |
| 20.1-a9 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{2} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$224.9714518$ |
0.770522476 |
\( -\frac{108768}{25} a^{2} + \frac{75424}{25} a + \frac{407296}{25} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - 3\) , \( 0\) , \( -625 a^{2} - 730 a + 290\) , \( -9146 a^{2} - 10702 a + 4214\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-625a^{2}-730a+290\right){x}-9146a^{2}-10702a+4214$ |
| 20.1-a10 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 5^{6} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2Cs, 3B.1.2 |
$9$ |
\( 2 \) |
$1$ |
$8.332275994$ |
0.770522476 |
\( \frac{7209817568672}{15625} a^{2} + \frac{8324376473504}{15625} a - \frac{3284389248384}{15625} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - 3\) , \( 0\) , \( -43485 a^{2} - 50880 a + 20040\) , \( -9168052 a^{2} - 10727414 a + 4224740\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(-43485a^{2}-50880a+20040\right){x}-9168052a^{2}-10727414a+4224740$ |
| 20.1-a11 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5^{3} \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.2 |
$9$ |
\( 1 \) |
$1$ |
$4.166137997$ |
0.770522476 |
\( \frac{3238786394956992}{125} a^{2} - \frac{8036058359040256}{125} a + \frac{2186604679552576}{125} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( -40 a^{2} + 102 a - 30\) , \( -285 a^{2} + 712 a - 203\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+\left(-40a^{2}+102a-30\right){x}-285a^{2}+712a-203$ |
| 20.1-a12 |
20.1-a |
$12$ |
$24$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 5 \) |
$1.79105$ |
$(a^2-a-2), (a^2-a-1)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$112.4857259$ |
0.770522476 |
\( \frac{43712}{5} a^{2} - \frac{105216}{5} a + \frac{28736}{5} \) |
\( \bigl[a^{2} - 1\) , \( a^{2} - a - 2\) , \( a^{2} - a - 2\) , \( 2 a\) , \( 2 a - 2\bigr] \) |
${y}^2+\left(a^{2}-1\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(a^{2}-a-2\right){x}^{2}+2a{x}+2a-2$ |
*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.
