| 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 |
| 5000.3-a1 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{10} \cdot 5^{22} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.866787005$ |
$0.299688898$ |
2.237821363 |
\( -\frac{35999730234}{390625} a - \frac{51700389912}{390625} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -751 i - 1082\) , \( 13990 i + 11750\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-751i-1082\right){x}+13990i+11750$ |
| 5000.3-a2 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{10} \cdot 5^{22} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1.866787005$ |
$0.299688898$ |
2.237821363 |
\( \frac{35999730234}{390625} a - \frac{51700389912}{390625} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( 750 i - 1082\) , \( -15072 i + 11000\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(750i-1082\right){x}-15072i+11000$ |
| 5000.3-a3 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{8} \cdot 5^{20} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.933393502$ |
$0.599377796$ |
2.237821363 |
\( \frac{237276}{625} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( -82\) , \( -572 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}-82{x}-572i$ |
| 5000.3-a4 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{11} \cdot 5^{29} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.733574011$ |
$0.149844449$ |
2.237821363 |
\( -\frac{22845545233191}{152587890625} a + \frac{135893651813613}{152587890625} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( 1500 i - 832\) , \( 3428 i + 25500\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(1500i-832\right){x}+3428i+25500$ |
| 5000.3-a5 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{11} \cdot 5^{29} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.733574011$ |
$0.149844449$ |
2.237821363 |
\( \frac{22845545233191}{152587890625} a + \frac{135893651813613}{152587890625} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -1501 i - 832\) , \( -4260 i + 27000\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-1501i-832\right){x}-4260i+27000$ |
| 5000.3-a6 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{4} \cdot 5^{16} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$0.466696751$ |
$1.198755592$ |
2.237821363 |
\( \frac{148176}{25} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -i + 43\) , \( 115 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+43\right){x}+115i$ |
| 5000.3-a7 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{8} \cdot 5^{14} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$0.233348375$ |
$1.198755592$ |
2.237821363 |
\( \frac{55296}{5} \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -50\) , \( 125\bigr] \) |
${y}^2={x}^{3}-50{x}+125$ |
| 5000.3-a8 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{8} \cdot 5^{14} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$0.933393502$ |
$0.599377796$ |
2.237821363 |
\( \frac{132304644}{5} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -i + 668\) , \( 6990 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i+668\right){x}+6990i$ |
| 5000.3-a9 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{11} \cdot 5^{17} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.733574011$ |
$0.149844449$ |
2.237821363 |
\( -\frac{15332659200009}{625} a + \frac{5763174879987}{625} \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( 12000 i - 17332\) , \( -937572 i + 718500\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+\left(12000i-17332\right){x}-937572i+718500$ |
| 5000.3-a10 |
5000.3-a |
$10$ |
$16$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{11} \cdot 5^{17} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.733574011$ |
$0.149844449$ |
2.237821363 |
\( \frac{15332659200009}{625} a + \frac{5763174879987}{625} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -12001 i - 17332\) , \( 920240 i + 730500\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-12001i-17332\right){x}+920240i+730500$ |
| 5000.3-b1 |
5000.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{8} \cdot 5^{6} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.073302756$ |
$3.843711527$ |
2.254037205 |
\( 2048 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -3\) , \( -2\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-3{x}-2$ |
| 5000.3-b2 |
5000.3-b |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{4} \cdot 5^{6} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.146605513$ |
$3.843711527$ |
2.254037205 |
\( 78608 \) |
\( \bigl[i + 1\) , \( -i\) , \( 0\) , \( 7\) , \( 6 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}-i{x}^{2}+7{x}+6i$ |
| 5000.3-c1 |
5000.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{8} \cdot 5^{18} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.768742305$ |
1.537484610 |
\( 2048 \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -83\) , \( 88\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-83{x}+88$ |
| 5000.3-c2 |
5000.3-c |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{4} \cdot 5^{18} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.768742305$ |
1.537484610 |
\( 78608 \) |
\( \bigl[i + 1\) , \( -i\) , \( i + 1\) , \( -i + 177\) , \( -927 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}-i{x}^{2}+\left(-i+177\right){x}-927i$ |
| 5000.3-d1 |
5000.3-d |
$1$ |
$1$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{10} \cdot 5^{4} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$0.128375062$ |
$4.573085028$ |
2.348280313 |
\( 270 \) |
\( \bigl[i + 1\) , \( i\) , \( 0\) , \( -2\) , \( -2 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}-2{x}-2i$ |
| 5000.3-e1 |
5000.3-e |
$1$ |
$1$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
5000.3 |
\( 2^{3} \cdot 5^{4} \) |
\( 2^{10} \cdot 5^{16} \) |
$1.50283$ |
$(a+1), (-a-2), (2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
✓ |
|
|
$1$ |
\( 2 \) |
$1$ |
$0.914617005$ |
1.829234011 |
\( 270 \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -i - 32\) , \( 140 i\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-i-32\right){x}+140i$ |
*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.
