| 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 |
| 45.1-a1 |
45.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{44} \cdot 5^{4} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.939955470$ |
$1.527514827$ |
3.208970777 |
\( -\frac{4173281}{1076168025} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 1879 a - 14369\) , \( -79147747 a + 606185522\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(w+1\right){x}^2+\left(1879w-14369\right){x}-79147747w+606185522$ |
| 45.1-a2 |
45.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{28} \cdot 5^{8} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.469977735$ |
$6.110059309$ |
3.208970777 |
\( \frac{233858751281}{4100625} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( 71879 a - 550494\) , \( -27767372 a + 212667822\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(w+1\right){x}^2+\left(71879w-550494\right){x}-27767372w+212667822$ |
| 45.1-b1 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{9} \) |
$2.706065934$ |
$0.490422220$ |
5.932142254 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 14186 a - 108628\) , \( 4712248 a - 36090667\bigr] \) |
${y}^2+w{x}{y}={x}^3+\left(-w+1\right){x}^2+\left(14186w-108628\right){x}+4712248w-36090667$ |
| 45.1-b2 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.706065934$ |
$31.38702211$ |
5.932142254 |
\( -\frac{1}{15} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( -4 a + 52\) , \( -1032 a + 7923\bigr] \) |
${y}^2+w{x}{y}={x}^3+\left(-w+1\right){x}^2+\left(-4w+52\right){x}-1032w+7923$ |
| 45.1-b3 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1.353032967$ |
$1.961688882$ |
5.932142254 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 4519 a + 30113\) , \( -206840 a - 1377310\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(4519w+30113\right){x}-206840w-1377310$ |
| 45.1-b4 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$0.676516483$ |
$7.846755528$ |
5.932142254 |
\( \frac{111284641}{50625} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -1286 a - 8542\) , \( -39368 a - 262129\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(-1286w-8542\right){x}-39368w-262129$ |
| 45.1-b5 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1.353032967$ |
$31.38702211$ |
5.932142254 |
\( \frac{13997521}{225} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 641 a - 4888\) , \( -19640 a + 150440\bigr] \) |
${y}^2+w{x}{y}={x}^3+\left(-w+1\right){x}^2+\left(641w-4888\right){x}-19640w+150440$ |
| 45.1-b6 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$1.353032967$ |
$1.961688882$ |
5.932142254 |
\( \frac{272223782641}{164025} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 17411 a - 133328\) , \( 3454968 a - 26461272\bigr] \) |
${y}^2+w{x}{y}={x}^3+\left(-w+1\right){x}^2+\left(17411w-133328\right){x}+3454968w-26461272$ |
| 45.1-b7 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$2.706065934$ |
$31.38702211$ |
5.932142254 |
\( \frac{56667352321}{15} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -10316 a - 68672\) , \( 1463000 a + 9742005\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(-10316w-68672\right){x}+1463000w+9742005$ |
| 45.1-b8 |
45.1-b |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$2.706065934$ |
$0.490422220$ |
5.932142254 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -278636 a - 1855392\) , \( -215960088 a - 1438058889\bigr] \) |
${y}^2+\left(w+1\right){x}{y}={x}^3+\left(-278636w-1855392\right){x}-215960088w-1438058889$ |
| 45.1-c1 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{32} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$4.300387515$ |
$0.490422220$ |
2.356789441 |
\( -\frac{147281603041}{215233605} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -110\) , \( -880\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-110{x}-880$ |
| 45.1-c2 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.300387515$ |
$31.38702211$ |
2.356789441 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2$ |
| 45.1-c3 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{16} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{6} \) |
$2.150193757$ |
$1.961688882$ |
2.356789441 |
\( \frac{4733169839}{3515625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 35\) , \( -28\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2+35{x}-28$ |
| 45.1-c4 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{8} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{5} \) |
$1.075096878$ |
$7.846755528$ |
2.356789441 |
\( \frac{111284641}{50625} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-10{x}-10$ |
| 45.1-c5 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.150193757$ |
$31.38702211$ |
2.356789441 |
\( \frac{13997521}{225} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -5\) , \( 2\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-5{x}+2$ |
| 45.1-c6 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{16} \cdot 5^{4} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$2.150193757$ |
$1.961688882$ |
2.356789441 |
\( \frac{272223782641}{164025} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -135\) , \( -660\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-135{x}-660$ |
| 45.1-c7 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.300387515$ |
$31.38702211$ |
2.356789441 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-80{x}+242$ |
| 45.1-c8 |
45.1-c |
$8$ |
$16$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{8} \cdot 5^{2} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$4.300387515$ |
$0.490422220$ |
2.356789441 |
\( \frac{1114544804970241}{405} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -2160\) , \( -39540\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-2160{x}-39540$ |
| 45.1-d1 |
45.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{44} \cdot 5^{4} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.939955470$ |
$1.527514827$ |
3.208970777 |
\( -\frac{4173281}{1076168025} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -1878 a - 12491\) , \( 79149624 a + 527050266\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(-w-1\right){x}^2+\left(-1878w-12491\right){x}+79149624w+527050266$ |
| 45.1-d2 |
45.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{28} \cdot 5^{8} \) |
$3.31374$ |
$(3,a), (3,a+2), (a+7)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.469977735$ |
$6.110059309$ |
3.208970777 |
\( \frac{233858751281}{4100625} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( -71878 a - 478616\) , \( 27839249 a + 185379066\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(-w-1\right){x}^2+\left(-71878w-478616\right){x}+27839249w+185379066$ |
*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.
