| 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 |
| 225.3-a1 |
225.3-a |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{3} \cdot 5^{4} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \cdot 3 \) |
$0.244702227$ |
$5.575121422$ |
3.341703234 |
\( 687 a - 1683 \) |
\( \bigl[a + 1\) , \( -1\) , \( a\) , \( -3\) , \( -2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}-{x}^{2}-3{x}-2$ |
| 225.3-b1 |
225.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( - 3^{9} \cdot 5^{11} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.334627598$ |
1.361356016 |
\( \frac{2537958}{3125} a + \frac{11450547}{3125} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -69 a + 166\) , \( -69 a + 168\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-69a+166\right){x}-69a+168$ |
| 225.3-b2 |
225.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{9} \cdot 5^{16} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$3.334627598$ |
1.361356016 |
\( \frac{19758072927}{9765625} a + \frac{65805407793}{9765625} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 276 a - 689\) , \( -759 a + 1878\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(276a-689\right){x}-759a+1878$ |
| 225.3-c1 |
225.3-c |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 3 \) |
$1$ |
$1.795632472$ |
1.099595830 |
\( -1835626496 a - 4496347136 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( -99 a + 243\) , \( -683 a + 1670\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-99a+243\right){x}-683a+1670$ |
| 225.3-c2 |
225.3-c |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{8} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 3 \) |
$1$ |
$1.795632472$ |
1.099595830 |
\( 118784 a - 290816 \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 61 a - 147\) , \( 455 a - 1116\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(61a-147\right){x}+455a-1116$ |
| 225.3-d1 |
225.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( - 3^{9} \cdot 5^{11} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.300946766$ |
0.877927082 |
\( \frac{2537958}{3125} a + \frac{11450547}{3125} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -54 a - 126\) , \( 204 a + 495\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-54a-126\right){x}+204a+495$ |
| 225.3-d2 |
225.3-d |
$2$ |
$2$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{9} \cdot 5^{16} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$2.150473383$ |
0.877927082 |
\( \frac{19758072927}{9765625} a + \frac{65805407793}{9765625} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( 0\) , \( -249 a - 621\) , \( -3531 a - 8640\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-249a-621\right){x}-3531a-8640$ |
| 225.3-e1 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.597533156$ |
$9.267456131$ |
2.260720761 |
\( 8000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 53 a - 126\) , \( 271 a - 662\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(53a-126\right){x}+271a-662$ |
| 225.3-e2 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-8$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 2^{2} \) |
$0.298766578$ |
$18.53491226$ |
2.260720761 |
\( 8000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -2 a - 6\) , \( 2\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-2a-6\right){x}+2$ |
| 225.3-e3 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.792599469$ |
$3.089152043$ |
2.260720761 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 4118 a - 10086\) , \( 223834 a - 548279\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(4118a-10086\right){x}+223834a-548279$ |
| 225.3-e4 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.896299734$ |
$6.178304087$ |
2.260720761 |
\( -77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -17 a - 246\) , \( -714 a - 547\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-17a-246\right){x}-714a-547$ |
| 225.3-e5 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$1.792599469$ |
$3.089152043$ |
2.260720761 |
\( 77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( 1028 a - 2526\) , \( -28451 a + 69661\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(1028a-2526\right){x}-28451a+69661$ |
| 225.3-e6 |
225.3-e |
$6$ |
$18$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{6} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{potential}$ |
$-72$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 2^{2} \) |
$0.896299734$ |
$6.178304087$ |
2.260720761 |
\( 77092288000 a + 188837384000 \) |
\( \bigl[a\) , \( a\) , \( 1\) , \( -227 a - 606\) , \( 2841 a + 6833\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-227a-606\right){x}+2841a+6833$ |
| 225.3-f1 |
225.3-f |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{9} \cdot 5^{10} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$2.540837879$ |
1.037292720 |
\( 687 a - 1683 \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 16 a - 41\) , \( 67 a - 162\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(16a-41\right){x}+67a-162$ |
| 225.3-g1 |
225.3-g |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{3} \cdot 5^{4} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.076360323$ |
$26.02867417$ |
1.622834300 |
\( 687 a - 1683 \) |
\( \bigl[1\) , \( -a - 1\) , \( 1\) , \( -a - 2\) , \( 147 a + 360\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-2\right){x}+147a+360$ |
| 225.3-h1 |
225.3-h |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$11.07119833$ |
2.259898897 |
\( -1835626496 a - 4496347136 \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -31 a - 69\) , \( 164 a + 403\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-31a-69\right){x}+164a+403$ |
| 225.3-h2 |
225.3-h |
$2$ |
$5$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{8} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$11.07119833$ |
2.259898897 |
\( 118784 a - 290816 \) |
\( \bigl[0\) , \( -a\) , \( a + 1\) , \( 9 a + 21\) , \( 61 a + 149\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(9a+21\right){x}+61a+149$ |
| 225.3-i1 |
225.3-i |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$5.475537501$ |
1.117689412 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( -6 a - 15\bigr] \) |
${y}^2+{y}={x}^{3}-6a-15$ |
| 225.3-i2 |
225.3-i |
$2$ |
$3$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{6} \cdot 5^{4} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 3 \) |
$1$ |
$16.42661250$ |
1.117689412 |
\( 0 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -11 a + 24\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-11a+24$ |
| 225.3-j1 |
225.3-j |
$1$ |
$1$ |
\(\Q(\sqrt{6}) \) |
$2$ |
$[2, 0]$ |
225.3 |
\( 3^{2} \cdot 5^{2} \) |
\( 3^{9} \cdot 5^{10} \) |
$1.69547$ |
$(a+3), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$3.807484666$ |
1.554399106 |
\( 687 a - 1683 \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -5 a - 12\) , \( -1005 a - 2463\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a-12\right){x}-1005a-2463$ |
*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.
