| 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 |
| 1.1-a1 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.57218$ |
$\textsf{none}$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$33.06583947$ |
0.322751033 |
\( -49130 a + 181781 \) |
\( \bigl[1\) , \( a\) , \( a\) , \( 3\) , \( -2\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+a{x}^{2}+3{x}-2$ |
| 1.1-a2 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.57218$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$8.266459867$ |
0.322751033 |
\( -10739384330 a + 39752499941 \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 40 a + 108\) , \( 10 a + 27\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(40a+108\right){x}+10a+27$ |
| 1.1-a3 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.57218$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$33.06583947$ |
0.322751033 |
\( -40800 a + 152753 \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -10 a - 27\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-10a-27\right){x}$ |
| 1.1-a4 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.57218$ |
$\textsf{none}$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2$ |
2Cs |
$1$ |
\( 1 \) |
$1$ |
$33.06583947$ |
0.322751033 |
\( 40800 a + 111953 \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 10 a - 37\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(10a-37\right){x}$ |
| 1.1-a5 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.57218$ |
$\textsf{none}$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$33.06583947$ |
0.322751033 |
\( 49130 a + 132651 \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -a + 3\) , \( -a - 2\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+3\right){x}-a-2$ |
| 1.1-a6 |
1.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( -1 \) |
$0.57218$ |
$\textsf{none}$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
✓ |
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$8.266459867$ |
0.322751033 |
\( 10739384330 a + 29013115611 \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( -40 a + 148\) , \( -10 a + 37\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-40a+148\right){x}-10a+37$ |
| 4.1-a1 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.414501108$ |
0.533255483 |
\( -\frac{12816149125}{4096} a + \frac{47424311875}{4096} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 635 a - 2350\) , \( 15849 a - 58666\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(635a-2350\right){x}+15849a-58666$ |
| 4.1-a2 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$30.73050997$ |
0.533255483 |
\( -\frac{1625}{16} a + \frac{28875}{16} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( 15 a - 55\) , \( -17 a + 63\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(15a-55\right){x}-17a+63$ |
| 4.1-a3 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{6} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$30.73050997$ |
0.533255483 |
\( \frac{1625}{16} a + \frac{13625}{8} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -15 a - 40\) , \( 17 a + 46\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-15a-40\right){x}+17a+46$ |
| 4.1-a4 |
4.1-a |
$4$ |
$6$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{18} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.414501108$ |
0.533255483 |
\( \frac{12816149125}{4096} a + \frac{17304081375}{2048} \) |
\( \bigl[1\) , \( 1\) , \( 0\) , \( -635 a - 1715\) , \( -15849 a - 42817\bigr] \) |
${y}^2+{x}{y}={x}^{3}+{x}^{2}+\left(-635a-1715\right){x}-15849a-42817$ |
| 4.1-b1 |
4.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{4} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.616775006$ |
1.541185169 |
\( -\frac{5770238117800857605}{2} a + \frac{42717789665650170789}{4} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -54427 a - 147034\) , \( -12468083 a - 33683302\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-54427a-147034\right){x}-12468083a-33683302$ |
| 4.1-b2 |
4.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{20} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$9.868400111$ |
1.541185169 |
\( -\frac{22041605}{65536} a + \frac{81575171}{65536} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 3 a - 9\) , \( -5\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(3a-9\right){x}-5$ |
| 4.1-b3 |
4.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{20} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$9.868400111$ |
1.541185169 |
\( \frac{22041605}{65536} a + \frac{29766783}{32768} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -4 a - 6\) , \( -a - 5\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-4a-6\right){x}-a-5$ |
| 4.1-b4 |
4.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{16} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{6} \) |
$1$ |
$9.868400111$ |
1.541185169 |
\( \frac{16974593}{256} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -3427 a - 9254\) , \( -188863 a - 510226\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-3427a-9254\right){x}-188863a-510226$ |
| 4.1-b5 |
4.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$2.467100027$ |
1.541185169 |
\( \frac{68769820673}{16} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -54627 a - 147574\) , \( -12372303 a - 33424546\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-54627a-147574\right){x}-12372303a-33424546$ |
| 4.1-b6 |
4.1-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( - 2^{4} \) |
$0.80918$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2^{2} \) |
$1$ |
$0.616775006$ |
1.541185169 |
\( \frac{5770238117800857605}{2} a + \frac{31177313430048455579}{4} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 54426 a - 201461\) , \( 12468082 a - 46151385\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(54426a-201461\right){x}+12468082a-46151385$ |
| 8.1-a1 |
8.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( - 2^{10} \) |
$0.96228$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$21.24582361$ |
1.659020099 |
\( \frac{1037}{4} a + \frac{3191}{2} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -141 a + 532\) , \( 1652 a - 6105\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-141a+532\right){x}+1652a-6105$ |
| 8.1-a2 |
8.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.1 |
\( 2^{3} \) |
\( 2^{5} \) |
$0.96228$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$42.49164722$ |
1.659020099 |
\( \frac{887825}{2} a + 1202403 \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 2 a + 4\) , \( a + 3\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a+4\right){x}+a+3$ |
| 8.2-a1 |
8.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( - 2^{10} \) |
$0.96228$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$21.24582361$ |
1.659020099 |
\( -\frac{1037}{4} a + \frac{7419}{4} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 147 a + 396\) , \( -1115 a - 3013\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(147a+396\right){x}-1115a-3013$ |
| 8.2-a2 |
8.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.2 |
\( 2^{3} \) |
\( 2^{5} \) |
$0.96228$ |
$(-a+4), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$42.49164722$ |
1.659020099 |
\( -\frac{887825}{2} a + \frac{3292631}{2} \) |
\( \bigl[a + 1\) , \( a\) , \( 0\) , \( 4 a + 11\) , \( 3 a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+a{x}^{2}+\left(4a+11\right){x}+3a+9$ |
| 8.3-a1 |
8.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{10} \) |
$0.96228$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.628169282$ |
$13.61564515$ |
1.335743261 |
\( -793 a + 2923 \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( 19 a - 53\) , \( 45 a - 143\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(19a-53\right){x}+45a-143$ |
| 8.3-a2 |
8.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.3 |
\( 2^{3} \) |
\( - 2^{8} \) |
$0.96228$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.314084641$ |
$13.61564515$ |
1.335743261 |
\( 19357 a + 52773 \) |
\( \bigl[a + 1\) , \( a\) , \( a + 1\) , \( -101 a - 273\) , \( -1337 a - 3613\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-101a-273\right){x}-1337a-3613$ |
| 8.4-a1 |
8.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{8} \) |
$0.96228$ |
$(-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.314084641$ |
$13.61564515$ |
1.335743261 |
\( -19357 a + 72130 \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( 105 a - 383\) , \( 958 a - 3541\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(105a-383\right){x}+958a-3541$ |
| 8.4-a2 |
8.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
8.4 |
\( 2^{3} \) |
\( - 2^{10} \) |
$0.96228$ |
$(-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$0.628169282$ |
$13.61564515$ |
1.335743261 |
\( 793 a + 2130 \) |
\( \bigl[a\) , \( a - 1\) , \( a\) , \( -15 a - 43\) , \( -84 a - 229\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-15a-43\right){x}-84a-229$ |
| 10.1-a1 |
10.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{8} \cdot 5^{2} \) |
$1.01749$ |
$(-a+4), (2a+5)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$26.54277691$ |
1.036321330 |
\( -\frac{40444573}{6400} a + \frac{151151403}{6400} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 4 a + 15\) , \( 17 a + 45\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(4a+15\right){x}+17a+45$ |
| 10.1-a2 |
10.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{16} \) |
$1.01749$ |
$(-a+4), (2a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.658923556$ |
1.036321330 |
\( \frac{63249956672063}{305175781250} a - \frac{160775740768643}{305175781250} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 32 a - 117\) , \( 253 a - 941\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(32a-117\right){x}+253a-941$ |
| 10.1-a3 |
10.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{8} \) |
$1.01749$ |
$(-a+4), (2a+5)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$6.635694227$ |
1.036321330 |
\( -\frac{723193137849}{1562500} a + \frac{2689218641939}{1562500} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 37 a - 137\) , \( 182 a - 679\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(37a-137\right){x}+182a-679$ |
| 10.1-a4 |
10.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.01749$ |
$(-a+4), (2a+5)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$26.54277691$ |
1.036321330 |
\( \frac{8680971}{10000} a + \frac{49344419}{10000} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 2 a - 7\) , \( -a - 1\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(2a-7\right){x}-a-1$ |
| 10.1-a5 |
10.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{4} \) |
$1.01749$ |
$(-a+4), (2a+5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.658923556$ |
1.036321330 |
\( -\frac{1722859110399887}{1250} a + \frac{6377332736960307}{1250} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( 602 a - 2237\) , \( 13763 a - 50969\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(602a-2237\right){x}+13763a-50969$ |
| 10.1-a6 |
10.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5^{2} \) |
$1.01749$ |
$(-a+4), (2a+5)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$26.54277691$ |
1.036321330 |
\( \frac{849817657}{100} a + \frac{2295947573}{100} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -1075 a + 3985\) , \( 25714 a - 95183\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1075a+3985\right){x}+25714a-95183$ |
| 10.2-a1 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{21} \cdot 5^{3} \) |
$1.01749$ |
$(-a+4), (2a-7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.854601199$ |
0.445813808 |
\( -\frac{215626570289}{262144000} a - \frac{79372877173}{52428800} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -306 a - 826\) , \( -7026 a - 18984\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-306a-826\right){x}-7026a-18984$ |
| 10.2-a2 |
10.2-a |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.2 |
\( 2 \cdot 5 \) |
\( 2^{7} \cdot 5 \) |
$1.01749$ |
$(-a+4), (2a-7)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25.69141079$ |
0.445813808 |
\( \frac{252071}{640} a + \frac{7363}{128} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 29 a + 79\) , \( 128 a + 343\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(29a+79\right){x}+128a+343$ |
| 10.3-a1 |
10.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.3 |
\( 2 \cdot 5 \) |
\( 2^{7} \cdot 5 \) |
$1.01749$ |
$(a+3), (2a+5)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$25.69141079$ |
0.445813808 |
\( -\frac{252071}{640} a + \frac{144443}{320} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( -24 a + 109\) , \( -49 a + 201\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-24a+109\right){x}-49a+201$ |
| 10.3-a2 |
10.3-a |
$2$ |
$3$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.3 |
\( 2 \cdot 5 \) |
\( 2^{21} \cdot 5^{3} \) |
$1.01749$ |
$(a+3), (2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.854601199$ |
0.445813808 |
\( \frac{215626570289}{262144000} a - \frac{306245478077}{131072000} \) |
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 311 a - 1131\) , \( 6200 a - 22930\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(311a-1131\right){x}+6200a-22930$ |
| 10.4-a1 |
10.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{16} \) |
$1.01749$ |
$(a+3), (2a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$1.658923556$ |
1.036321330 |
\( -\frac{63249956672063}{305175781250} a - \frac{9752578409658}{30517578125} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -33 a - 84\) , \( -254 a - 687\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-33a-84\right){x}-254a-687$ |
| 10.4-a2 |
10.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( - 2^{2} \cdot 5^{2} \) |
$1.01749$ |
$(a+3), (2a-7)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$26.54277691$ |
1.036321330 |
\( -\frac{849817657}{100} a + \frac{314576523}{10} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 1076 a + 2909\) , \( -26791 a - 72378\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(1076a+2909\right){x}-26791a-72378$ |
| 10.4-a3 |
10.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{4} \cdot 5^{4} \) |
$1.01749$ |
$(a+3), (2a-7)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$26.54277691$ |
1.036321330 |
\( -\frac{8680971}{10000} a + \frac{5802539}{1000} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -3 a - 4\) , \( -1\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-3a-4\right){x}-1$ |
| 10.4-a4 |
10.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( - 2^{8} \cdot 5^{2} \) |
$1.01749$ |
$(a+3), (2a-7)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$26.54277691$ |
1.036321330 |
\( \frac{40444573}{6400} a + \frac{11070683}{640} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -5 a + 19\) , \( -18 a + 62\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-5a+19\right){x}-18a+62$ |
| 10.4-a5 |
10.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2^{2} \cdot 5^{8} \) |
$1.01749$ |
$(a+3), (2a-7)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$6.635694227$ |
1.036321330 |
\( \frac{723193137849}{1562500} a + \frac{196602550409}{156250} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -38 a - 99\) , \( -183 a - 496\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-38a-99\right){x}-183a-496$ |
| 10.4-a6 |
10.4-a |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
10.4 |
\( 2 \cdot 5 \) |
\( 2 \cdot 5^{4} \) |
$1.01749$ |
$(a+3), (2a-7)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$1.658923556$ |
1.036321330 |
\( \frac{1722859110399887}{1250} a + \frac{465447362656042}{125} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( -603 a - 1634\) , \( -13764 a - 37205\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-603a-1634\right){x}-13764a-37205$ |
| 16.2-a1 |
16.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{18} \) |
$1.14435$ |
$(-a+4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.131947458$ |
$26.49421336$ |
1.091918255 |
\( -\frac{1584117}{256} a + \frac{2923155}{256} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( -104 a - 280\) , \( 912 a + 2464\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(-104a-280\right){x}+912a+2464$ |
| 16.2-a2 |
16.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.2 |
\( 2^{4} \) |
\( - 2^{12} \) |
$1.14435$ |
$(-a+4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.065973729$ |
$26.49421336$ |
1.091918255 |
\( \frac{372808413}{16} a + \frac{1007165205}{16} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( a + 5\) , \( -a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(a+5\right){x}-a+7$ |
| 16.3-a1 |
16.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{12} \) |
$1.14435$ |
$(-a+4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.065973729$ |
$26.49421336$ |
1.091918255 |
\( -\frac{372808413}{16} a + \frac{689986809}{8} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -5\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}-5{x}+a+1$ |
| 16.3-a2 |
16.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.3 |
\( 2^{4} \) |
\( - 2^{18} \) |
$1.14435$ |
$(-a+4), (a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.131947458$ |
$26.49421336$ |
1.091918255 |
\( \frac{1584117}{256} a + \frac{669519}{128} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( 105 a - 395\) , \( -1017 a + 3761\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(105a-395\right){x}-1017a+3761$ |
| 16.4-a1 |
16.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( - 2^{8} \) |
$1.14435$ |
$(-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.40899837$ |
1.984109430 |
\( -19357 a + 72130 \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 4 a + 7\) , \( 4 a + 9\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+7\right){x}+4a+9$ |
| 16.4-a2 |
16.4-a |
$2$ |
$2$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( - 2^{10} \) |
$1.14435$ |
$(-a+4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$25.40899837$ |
1.984109430 |
\( 793 a + 2130 \) |
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( -a - 3\) , \( -12 a + 41\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-3\right){x}-12a+41$ |
| 16.4-b1 |
16.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( - 2^{12} \) |
$1.14435$ |
$(-a+4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.379505841$ |
$8.207659751$ |
0.972917189 |
\( -49130 a + 181781 \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -2 a - 9\) , \( -13 a - 37\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-2a-9\right){x}-13a-37$ |
| 16.4-b2 |
16.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( - 2^{12} \) |
$1.14435$ |
$(-a+4)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.759011682$ |
$16.41531950$ |
0.972917189 |
\( -10739384330 a + 39752499941 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 1300 a + 3511\) , \( 3153 a + 8517\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(1300a+3511\right){x}+3153a+8517$ |
| 16.4-b3 |
16.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.14435$ |
$(-a+4)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.518023365$ |
$32.83063900$ |
0.972917189 |
\( -40800 a + 152753 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -325 a - 879\) , \( -325 a - 879\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-325a-879\right){x}-325a-879$ |
| 16.4-b4 |
16.4-b |
$6$ |
$8$ |
\(\Q(\sqrt{41}) \) |
$2$ |
$[2, 0]$ |
16.4 |
\( 2^{4} \) |
\( 2^{12} \) |
$1.14435$ |
$(-a+4)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
✓ |
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.759011682$ |
$16.41531950$ |
0.972917189 |
\( 40800 a + 111953 \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 5 a - 19\) , \( 5 a - 19\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(5a-19\right){x}+5a-19$ |
*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.
