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 |
72.2-a1 |
72.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
72.2 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$0.73624$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$1$ |
$1.817673508$ |
0.642644632 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 5\) , \( 23\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+5{x}+23$ |
144.2-a1 |
144.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
144.2 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$0.87554$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$1$ |
$1.817673508$ |
1.285289264 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 5\) , \( -22\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+5{x}-22$ |
648.3-a1 |
648.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
648.3 |
\( 2^{3} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{28} \) |
$1.27520$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.079273864$ |
$0.605891169$ |
1.849572148 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 37\) , \( -607\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+37{x}-607$ |
1296.3-b1 |
1296.3-b |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
1296.3 |
\( 2^{4} \cdot 3^{4} \) |
\( 2^{10} \cdot 3^{28} \) |
$1.51647$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.605891169$ |
1.713719018 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -1\) , \( 0\) , \( 36\) , \( 572\bigr] \) |
${y}^2+a{x}{y}={x}^{3}-{x}^{2}+36{x}+572$ |
6912.2-a1 |
6912.2-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{22} \cdot 3^{22} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.859743848$ |
$0.524717144$ |
2.760090861 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 32 a - 16\) , \( 180 a - 900\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(32a-16\right){x}+180a-900$ |
6912.2-n1 |
6912.2-n |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.2 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{22} \cdot 3^{22} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.524717144$ |
2.968248410 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 32 a - 16\) , \( -180 a + 900\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(32a-16\right){x}-180a+900$ |
6912.3-a1 |
6912.3-a |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{22} \cdot 3^{22} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.859743848$ |
$0.524717144$ |
2.760090861 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -32 a - 16\) , \( -180 a - 900\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-32a-16\right){x}-180a-900$ |
6912.3-n1 |
6912.3-n |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6912.3 |
\( 2^{8} \cdot 3^{3} \) |
\( 2^{22} \cdot 3^{22} \) |
$2.30454$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.524717144$ |
2.968248410 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( -32 a - 16\) , \( 180 a + 900\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(-32a-16\right){x}+180a+900$ |
9216.2-m1 |
9216.2-m |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{16} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.642644632$ |
1.817673508 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -32\) , \( -360 a\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}-32{x}-360a$ |
9216.2-o1 |
9216.2-o |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
9216.2 |
\( 2^{10} \cdot 3^{2} \) |
\( 2^{28} \cdot 3^{16} \) |
$2.47639$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$0.642644632$ |
1.817673508 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( -32\) , \( 360 a\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}-32{x}+360a$ |
20808.4-c1 |
20808.4-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20808.4 |
\( 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{10} \cdot 3^{16} \cdot 17^{6} \) |
$3.03558$ |
$(a), (-a-1), (a-1), (-2a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.440850580$ |
2.493827480 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -46 a + 4\) , \( -808 a - 1016\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-46a+4\right){x}-808a-1016$ |
20808.6-c1 |
20808.6-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
20808.6 |
\( 2^{3} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{10} \cdot 3^{16} \cdot 17^{6} \) |
$3.03558$ |
$(a), (-a-1), (a-1), (2a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.440850580$ |
2.493827480 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 46 a + 4\) , \( 808 a - 1016\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(46a+4\right){x}+808a-1016$ |
26136.4-d1 |
26136.4-d |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.4 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{22} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.766583000$ |
$0.316416343$ |
5.488492471 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 31 a - 121\) , \( -2488 a - 2074\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(31a-121\right){x}-2488a-2074$ |
26136.6-i1 |
26136.6-i |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.6 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{22} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a-3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$1$ |
$0.316416343$ |
3.579842277 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -a + 1\) , \( 0\) , \( 78 a + 66\) , \( -2936 a - 46\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(78a+66\right){x}-2936a-46$ |
26136.7-i1 |
26136.7-i |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.7 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{22} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$1$ |
$0.316416343$ |
3.579842277 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( -78 a + 66\) , \( 2936 a - 46\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-78a+66\right){x}+2936a-46$ |
26136.9-d1 |
26136.9-d |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
26136.9 |
\( 2^{3} \cdot 3^{3} \cdot 11^{2} \) |
\( 2^{10} \cdot 3^{22} \cdot 11^{6} \) |
$3.21361$ |
$(a), (-a-1), (a-1), (a-3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.766583000$ |
$0.316416343$ |
5.488492471 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -31 a - 121\) , \( 2488 a - 2074\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-31a-121\right){x}+2488a-2074$ |
27648.2-v1 |
27648.2-v |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{28} \cdot 3^{22} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.030505696$ |
$0.371031051$ |
4.229750882 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -62 a + 31\) , \( 1737 a + 751\bigr] \) |
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-62a+31\right){x}+1737a+751$ |
27648.2-bb1 |
27648.2-bb |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.2 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{28} \cdot 3^{22} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.371031051$ |
2.098868579 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -62 a + 31\) , \( -1737 a - 751\bigr] \) |
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-62a+31\right){x}-1737a-751$ |
27648.3-f1 |
27648.3-f |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{28} \cdot 3^{22} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$4.030505696$ |
$0.371031051$ |
4.229750882 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 62 a + 31\) , \( -1737 a + 751\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(62a+31\right){x}-1737a+751$ |
27648.3-bp1 |
27648.3-bp |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
27648.3 |
\( 2^{10} \cdot 3^{3} \) |
\( 2^{28} \cdot 3^{22} \) |
$3.25911$ |
$(a), (-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{7} \) |
$1$ |
$0.371031051$ |
2.098868579 |
\( \frac{207646}{6561} \) |
\( \bigl[0\) , \( a - 1\) , \( 0\) , \( 62 a + 31\) , \( 1737 a - 751\bigr] \) |
${y}^2={x}^{3}+\left(a-1\right){x}^{2}+\left(62a+31\right){x}+1737a-751$ |
41616.4-c1 |
41616.4-c |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
41616.4 |
\( 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{10} \cdot 3^{16} \cdot 17^{6} \) |
$3.60993$ |
$(a), (-a-1), (a-1), (-2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.518025393$ |
$0.440850580$ |
5.167463847 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -48 a + 5\) , \( 855 a + 1013\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-48a+5\right){x}+855a+1013$ |
41616.6-g1 |
41616.6-g |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
41616.6 |
\( 2^{4} \cdot 3^{2} \cdot 17^{2} \) |
\( 2^{10} \cdot 3^{16} \cdot 17^{6} \) |
$3.60993$ |
$(a), (-a-1), (a-1), (2a+3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{8} \) |
$0.518025393$ |
$0.440850580$ |
5.167463847 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 48 a + 5\) , \( -855 a + 1013\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(48a+5\right){x}-855a+1013$ |
45000.2-i1 |
45000.2-i |
$8$ |
$16$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
45000.2 |
\( 2^{3} \cdot 3^{2} \cdot 5^{4} \) |
\( 2^{10} \cdot 3^{16} \cdot 5^{12} \) |
$3.68118$ |
$(a), (-a-1), (a-1), (5)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{9} \) |
$0.534270860$ |
$0.363534701$ |
8.789665301 |
\( \frac{207646}{6561} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 98\) , \( 2714\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+98{x}+2714$ |
*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.