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 |
72.1-a4 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$0.52060$ |
$(a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$3.635347017$ |
0.454418377 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -i + 6\) , \( -5 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i+6\right){x}-5i$ |
648.1-a4 |
648.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
648.1 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{8} \cdot 3^{20} \) |
$0.90170$ |
$(a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.211782339$ |
1.211782339 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -i + 54\) , \( -122 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+54\right){x}-122i$ |
2304.1-c4 |
2304.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
2304.1 |
\( 2^{8} \cdot 3^{2} \) |
\( 2^{20} \cdot 3^{8} \) |
$1.23820$ |
$(a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.817673508$ |
1.817673508 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( i\) , \( 0\) , \( 24\) , \( -36 i\bigr] \) |
${y}^2={x}^{3}+i{x}^{2}+24{x}-36i$ |
3600.1-c4 |
3600.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
3600.1 |
\( 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 5^{6} \) |
$1.38434$ |
$(a+1), (-a-2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.625776610$ |
1.625776610 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( 1\) , \( i + 1\) , \( -25 i - 18\) , \( 49 i + 9\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+{x}^{2}+\left(-25i-18\right){x}+49i+9$ |
3600.3-c4 |
3600.3-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
3600.3 |
\( 2^{4} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 5^{6} \) |
$1.38434$ |
$(a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$1.625776610$ |
1.625776610 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( -i + 1\) , \( i + 1\) , \( 23 i - 18\) , \( -50 i + 9\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(23i-18\right){x}-50i+9$ |
9216.1-c4 |
9216.1-c |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
9216.1 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{26} \cdot 3^{8} \) |
$1.75107$ |
$(a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.141888207$ |
$1.285289264$ |
2.752945917 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -i - 1\) , \( 0\) , \( -48 i\) , \( -72 i + 72\bigr] \) |
${y}^2={x}^{3}+\left(-i-1\right){x}^{2}-48i{x}-72i+72$ |
9216.1-j4 |
9216.1-j |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
9216.1 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{26} \cdot 3^{8} \) |
$1.75107$ |
$(a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$2.141888207$ |
$1.285289264$ |
2.752945917 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( i - 1\) , \( 0\) , \( 48 i\) , \( 72 i + 72\bigr] \) |
${y}^2={x}^{3}+\left(i-1\right){x}^{2}+48i{x}+72i+72$ |
20736.1-b4 |
20736.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
20736.1 |
\( 2^{8} \cdot 3^{4} \) |
\( 2^{20} \cdot 3^{20} \) |
$2.14462$ |
$(a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$2.158547729$ |
$0.605891169$ |
2.615690016 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 219\) , \( -1190 i\bigr] \) |
${y}^2={x}^{3}+219{x}-1190i$ |
20808.1-a4 |
20808.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
20808.1 |
\( 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 17^{6} \) |
$2.14648$ |
$(a+1), (a+4), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.881701161$ |
1.763402322 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( -i + 1\) , \( 0\) , \( -49 i - 91\) , \( -260 i - 325\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(-49i-91\right){x}-260i-325$ |
20808.3-a4 |
20808.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
20808.3 |
\( 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 17^{6} \) |
$2.14648$ |
$(a+1), (a-4), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.881701161$ |
1.763402322 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( 1\) , \( 0\) , \( 49 i - 91\) , \( 260 i - 325\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+{x}^{2}+\left(49i-91\right){x}+260i-325$ |
24336.1-a4 |
24336.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
24336.1 |
\( 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 13^{6} \) |
$2.23219$ |
$(a+1), (-3a-2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.899714244$ |
$1.008263852$ |
3.628597400 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( i - 1\) , \( 0\) , \( -74 i + 30\) , \( -10 i + 280\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i-1\right){x}^{2}+\left(-74i+30\right){x}-10i+280$ |
24336.3-a4 |
24336.3-a |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
24336.3 |
\( 2^{4} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{8} \cdot 13^{6} \) |
$2.23219$ |
$(a+1), (2a+3), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.899714244$ |
$1.008263852$ |
3.628597400 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( 0\) , \( 74 i + 30\) , \( -10 i - 280\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(74i+30\right){x}-10i-280$ |
32400.1-b4 |
32400.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
32400.1 |
\( 2^{4} \cdot 3^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 3^{20} \cdot 5^{6} \) |
$2.39775$ |
$(a+1), (-a-2), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.541925536$ |
1.083851073 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( -219 i - 165\) , \( 1554 i + 407\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(-219i-165\right){x}+1554i+407$ |
32400.3-b4 |
32400.3-b |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
32400.3 |
\( 2^{4} \cdot 3^{4} \cdot 5^{2} \) |
\( 2^{8} \cdot 3^{20} \cdot 5^{6} \) |
$2.39775$ |
$(a+1), (2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.541925536$ |
1.083851073 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( 219 i - 165\) , \( 1554 i - 407\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(219i-165\right){x}+1554i-407$ |
45000.3-k4 |
45000.3-k |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
45000.3 |
\( 2^{3} \cdot 3^{2} \cdot 5^{4} \) |
\( 2^{8} \cdot 3^{8} \cdot 5^{12} \) |
$2.60299$ |
$(a+1), (-a-2), (2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.187120860$ |
$0.727069403$ |
4.353595283 |
\( \frac{1556068}{81} \) |
\( \bigl[i + 1\) , \( -i\) , \( 0\) , \( 152\) , \( -714 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}-i{x}^{2}+152{x}-714i$ |
57600.1-d4 |
57600.1-d |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
57600.1 |
\( 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{20} \cdot 3^{8} \cdot 5^{6} \) |
$2.76869$ |
$(a+1), (-a-2), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.062418350$ |
$0.812888305$ |
3.454509811 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( i - 1\) , \( 0\) , \( -98 i - 73\) , \( 493 i + 145\bigr] \) |
${y}^2={x}^{3}+\left(i-1\right){x}^{2}+\left(-98i-73\right){x}+493i+145$ |
57600.3-d4 |
57600.3-d |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
57600.3 |
\( 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
\( 2^{20} \cdot 3^{8} \cdot 5^{6} \) |
$2.76869$ |
$(a+1), (2a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.062418350$ |
$0.812888305$ |
3.454509811 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( -i - 1\) , \( 0\) , \( 98 i - 73\) , \( -493 i + 145\bigr] \) |
${y}^2={x}^{3}+\left(-i-1\right){x}^{2}+\left(98i-73\right){x}-493i+145$ |
82944.1-g4 |
82944.1-g |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
82944.1 |
\( 2^{10} \cdot 3^{4} \) |
\( 2^{26} \cdot 3^{20} \) |
$3.03295$ |
$(a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.550999615$ |
$0.428429754$ |
4.756426807 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( 438 i\) , \( 2380 i + 2380\bigr] \) |
${y}^2={x}^{3}+438i{x}+2380i+2380$ |
82944.1-n4 |
82944.1-n |
$6$ |
$8$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
82944.1 |
\( 2^{10} \cdot 3^{4} \) |
\( 2^{26} \cdot 3^{20} \) |
$3.03295$ |
$(a+1), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.550999615$ |
$0.428429754$ |
4.756426807 |
\( \frac{1556068}{81} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -438 i\) , \( -2380 i + 2380\bigr] \) |
${y}^2={x}^{3}-438i{x}-2380i+2380$ |
*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.