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 |
900.1-a1 |
900.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{46} \cdot 3^{22} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
|
|
$169$ |
\( 2 \) |
$1$ |
$0.096436813$ |
4.477355893 |
\( \frac{657300262000123}{37150418534400} \) |
\( \bigl[a\) , \( 1\) , \( 1\) , \( 12680 a + 39857\) , \( -14564747 a - 45734865\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+{x}^{2}+\left(12680a+39857\right){x}-14564747a-45734865$ |
900.1-b1 |
900.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{44} \cdot 3^{16} \cdot 5^{18} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$289$ |
\( 2^{2} \) |
$1$ |
$0.094856309$ |
3.765530159 |
\( \frac{97802241300184795037}{53747712000000000} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -671902 a - 2111688\) , \( 127257712 a + 399601092\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-671902a-2111688\right){x}+127257712a+399601092$ |
900.1-b2 |
900.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{22} \cdot 3^{8} \cdot 5^{36} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1156$ |
\( 2^{2} \) |
$1$ |
$0.023714077$ |
3.765530159 |
\( \frac{87649400407713844299677}{632812500000000000} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -6477982 a - 20359368\) , \( -17036841872 a - 53497575996\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}+\left(-6477982a-20359368\right){x}-17036841872a-53497575996$ |
900.1-c1 |
900.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.879573008$ |
1.032716833 |
\( -\frac{30261551896446437}{150} a + \frac{375853462684920157}{450} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 674 a - 2776\) , \( 17951 a - 74315\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(674a-2776\right){x}+17951a-74315$ |
900.1-c2 |
900.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 3^{8} \cdot 5^{2} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.518292035$ |
1.032716833 |
\( -\frac{2801426243}{540} a + \frac{17847958663}{810} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 44 a - 166\) , \( 275 a - 1127\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(44a-166\right){x}+275a-1127$ |
900.1-d1 |
900.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.08846273$ |
2.072559750 |
\( \frac{79507}{300} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 9 a + 30\) , \( 59 a + 185\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(9a+30\right){x}+59a+185$ |
900.1-d2 |
900.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 5^{8} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$15.08846273$ |
2.072559750 |
\( \frac{83453453}{11250} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -61 a - 190\) , \( 287 a + 901\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-61a-190\right){x}+287a+901$ |
900.1-e1 |
900.1-e |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{14} \cdot 3^{10} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$3.976308653$ |
1.092375998 |
\( \frac{2712805671127662691}{777600} a - \frac{11231164533693007217}{777600} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -510 a - 1899\) , \( 17937 a + 58401\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-510a-1899\right){x}+17937a+58401$ |
900.1-f1 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{24} \cdot 3^{2} \cdot 5^{6} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1.544707650$ |
$1.248395236$ |
4.767964034 |
\( -\frac{273359449}{1536000} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -14\) , \( -64\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-14{x}-64$ |
900.1-f2 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{8} \cdot 3^{6} \cdot 5^{2} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$4.634122951$ |
$11.23555713$ |
4.767964034 |
\( \frac{357911}{2160} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 1\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}+2$ |
900.1-f3 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{6} \cdot 3^{2} \cdot 5^{24} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$1.544707650$ |
$1.248395236$ |
4.767964034 |
\( \frac{10316097499609}{5859375000} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -454\) , \( -544\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-454{x}-544$ |
900.1-f4 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{24} \cdot 5^{2} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$18.53649180$ |
$2.808889283$ |
4.767964034 |
\( \frac{35578826569}{5314410} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -69\) , \( -194\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-69{x}-194$ |
900.1-f5 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 3^{12} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{3} \cdot 3 \) |
$9.268245903$ |
$11.23555713$ |
4.767964034 |
\( \frac{702595369}{72900} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -19\) , \( 26\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-19{x}+26$ |
900.1-f6 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{12} \cdot 3^{4} \cdot 5^{12} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{3} \cdot 3^{2} \) |
$3.089415301$ |
$1.248395236$ |
4.767964034 |
\( \frac{4102915888729}{9000000} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -334\) , \( -2368\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-334{x}-2368$ |
900.1-f7 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{6} \cdot 5^{8} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3 \) |
$4.634122951$ |
$11.23555713$ |
4.767964034 |
\( \frac{2656166199049}{33750} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -289\) , \( 1862\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-289{x}+1862$ |
900.1-f8 |
900.1-f |
$8$ |
$12$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{6} \cdot 3^{8} \cdot 5^{6} \) |
$3.56317$ |
$(2), (3), (5)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3^{2} \) |
$6.178830602$ |
$0.312098809$ |
4.767964034 |
\( \frac{16778985534208729}{81000} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -5334\) , \( -150368\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-5334{x}-150368$ |
900.1-g1 |
900.1-g |
$1$ |
$1$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{14} \cdot 3^{10} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$1$ |
$3.976308653$ |
1.092375998 |
\( -\frac{2712805671127662691}{777600} a - \frac{4259179431282672263}{388800} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 519 a - 2425\) , \( -20361 a + 85460\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(519a-2425\right){x}-20361a+85460$ |
900.1-h1 |
900.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{4} \cdot 3^{8} \cdot 5^{2} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$7.518292035$ |
1.032716833 |
\( \frac{2801426243}{540} a + \frac{27291638597}{1620} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -35 a - 115\) , \( -440 a - 1375\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-35a-115\right){x}-440a-1375$ |
900.1-h2 |
900.1-h |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 3^{4} \cdot 5^{4} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.879573008$ |
1.032716833 |
\( \frac{30261551896446437}{150} a + \frac{142534403497790423}{225} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -665 a - 2095\) , \( -20726 a - 65077\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-665a-2095\right){x}-20726a-65077$ |
900.1-i1 |
900.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{8} \cdot 3^{2} \cdot 5^{20} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$0.929939837$ |
0.510948242 |
\( \frac{769330693747}{468750000} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( -1330 a + 5556\) , \( 12124 a - 50120\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1330a+5556\right){x}+12124a-50120$ |
900.1-i2 |
900.1-i |
$2$ |
$2$ |
\(\Q(\sqrt{53}) \) |
$2$ |
$[2, 0]$ |
900.1 |
\( 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{16} \cdot 3^{4} \cdot 5^{10} \) |
$3.56317$ |
$(2), (3), (5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.719759351$ |
0.510948242 |
\( \frac{13094193293}{7200000} \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -341 a - 1070\) , \( -2441 a - 7665\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-341a-1070\right){x}-2441a-7665$ |
*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.