| 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 |
| 4.1-a1 |
4.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{16} \cdot 3^{12} \) |
$1.80938$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.321671129$ |
$30.74727065$ |
2.763132533 |
\( \frac{16974593}{256} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -2999 a - 19949\) , \( 229075 a + 1525367\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(-2999w-19949\right){x}+229075w+1525367$ |
| 4.1-a2 |
4.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.80938$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.643342259$ |
$30.74727065$ |
2.763132533 |
\( \frac{68769820673}{16} \) |
\( \bigl[1\) , \( a - 1\) , \( a + 1\) , \( -47799 a - 318269\) , \( 15109635 a + 100613687\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(w-1\right){x}^2+\left(-47799w-318269\right){x}+15109635w+100613687$ |
| 4.1-b1 |
4.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{16} \cdot 3^{12} \) |
$1.80938$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.321671129$ |
$30.74727065$ |
2.763132533 |
\( \frac{16974593}{256} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 2998 a - 22947\) , \( -229076 a + 1754443\bigr] \) |
${y}^2+{x}{y}+w{y}={x}^3-w{x}^2+\left(2998w-22947\right){x}-229076w+1754443$ |
| 4.1-b2 |
4.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
4.1 |
\( 2^{2} \) |
\( 2^{8} \cdot 3^{12} \) |
$1.80938$ |
$(2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.643342259$ |
$30.74727065$ |
2.763132533 |
\( \frac{68769820673}{16} \) |
\( \bigl[1\) , \( -a\) , \( a\) , \( 47798 a - 366067\) , \( -15109636 a + 115723323\bigr] \) |
${y}^2+{x}{y}+w{y}={x}^3-w{x}^2+\left(47798w-366067\right){x}-15109636w+115723323$ |
| 12.1-a1 |
12.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{2} \cdot 3^{8} \) |
$2.38128$ |
$(3,a), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$27.17693563$ |
3.796239038 |
\( -\frac{327111324371}{13122} a + \frac{2505380625335}{13122} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 44 a - 156\) , \( -174 a + 1800\bigr] \) |
${y}^2+w{x}{y}={x}^3+\left(w+1\right){x}^2+\left(44w-156\right){x}-174w+1800$ |
| 12.1-a2 |
12.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.38128$ |
$(3,a), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$27.17693563$ |
3.796239038 |
\( -\frac{127985}{324} a + \frac{1378343}{81} \) |
\( \bigl[a\) , \( a + 1\) , \( 0\) , \( 14 a + 74\) , \( 32 a + 224\bigr] \) |
${y}^2+w{x}{y}={x}^3+\left(w+1\right){x}^2+\left(14w+74\right){x}+32w+224$ |
| 12.1-b1 |
12.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{2} \cdot 3^{8} \) |
$2.38128$ |
$(3,a), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.754871855$ |
$27.17693563$ |
3.330956522 |
\( -\frac{327111324371}{13122} a + \frac{2505380625335}{13122} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -43 a - 274\) , \( 210 a + 1385\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(w+1\right){x}^2+\left(-43w-274\right){x}+210w+1385$ |
| 12.1-b2 |
12.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.1 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.38128$ |
$(3,a), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.877435927$ |
$27.17693563$ |
3.330956522 |
\( -\frac{127985}{324} a + \frac{1378343}{81} \) |
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -13 a - 74\) , \( -108 a - 733\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(w+1\right){x}^2+\left(-13w-74\right){x}-108w-733$ |
| 12.2-a1 |
12.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.38128$ |
$(3,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$27.17693563$ |
3.796239038 |
\( \frac{127985}{324} a + \frac{1795129}{108} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( 12 a + 61\) , \( 29 a + 182\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+w{y}={x}^3+\left(w-1\right){x}^2+\left(12w+61\right){x}+29w+182$ |
| 12.2-a2 |
12.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{2} \cdot 3^{8} \) |
$2.38128$ |
$(3,a+2), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$27.17693563$ |
3.796239038 |
\( \frac{327111324371}{13122} a + \frac{363044883494}{2187} \) |
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -18 a - 139\) , \( 35 a + 222\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+w{y}={x}^3+\left(w-1\right){x}^2+\left(-18w-139\right){x}+35w+222$ |
| 12.2-b1 |
12.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{4} \cdot 3^{4} \) |
$2.38128$ |
$(3,a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.877435927$ |
$27.17693563$ |
3.330956522 |
\( \frac{127985}{324} a + \frac{1795129}{108} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 14 a - 88\) , \( 93 a - 753\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(-w-1\right){x}^2+\left(14w-88\right){x}+93w-753$ |
| 12.2-b2 |
12.2-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{2} \cdot 3^{8} \) |
$2.38128$ |
$(3,a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.754871855$ |
$27.17693563$ |
3.330956522 |
\( \frac{327111324371}{13122} a + \frac{363044883494}{2187} \) |
\( \bigl[1\) , \( -a - 1\) , \( a + 1\) , \( 44 a - 318\) , \( -255 a + 1913\bigr] \) |
${y}^2+{x}{y}+\left(w+1\right){y}={x}^3+\left(-w-1\right){x}^2+\left(44w-318\right){x}-255w+1913$ |
| 20.1-a1 |
20.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 3^{12} \cdot 5^{16} \) |
$2.70566$ |
$(a+7), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1.693362080$ |
$4.029590656$ |
3.812622599 |
\( \frac{63521199}{1562500} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 4655 a + 30997\) , \( 2978860 a + 19835962\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(4655w+30997\right){x}+2978860w+19835962$ |
| 20.1-a2 |
20.1-a |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 3^{12} \cdot 5^{8} \) |
$2.70566$ |
$(a+7), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.846681040$ |
$16.11836262$ |
3.812622599 |
\( \frac{176558481}{10000} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -6545 a - 43583\) , \( 734100 a + 4888306\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-6545w-43583\right){x}+734100w+4888306$ |
| 20.1-b1 |
20.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{4} \cdot 3^{12} \cdot 5^{16} \) |
$2.70566$ |
$(a+7), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1.693362080$ |
$4.029590656$ |
3.812622599 |
\( \frac{63521199}{1562500} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -4655 a + 35652\) , \( -2978860 a + 22814822\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-4655w+35652\right){x}-2978860w+22814822$ |
| 20.1-b2 |
20.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{8} \cdot 3^{12} \cdot 5^{8} \) |
$2.70566$ |
$(a+7), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.846681040$ |
$16.11836262$ |
3.812622599 |
\( \frac{176558481}{10000} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 6545 a - 50128\) , \( -734100 a + 5622406\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(6545w-50128\right){x}-734100w+5622406$ |
| 27.1-a1 |
27.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$7.911270077$ |
$0.944344642$ |
2.087179461 |
\( -\frac{169467576320}{243} a - \frac{376156487680}{81} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -391 a - 2589\) , \( -10278 a - 68430\bigr] \) |
${y}^2+{y}={x}^3-w{x}^2+\left(-391w-2589\right){x}-10278w-68430$ |
| 27.1-a2 |
27.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{34} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$2.637090025$ |
$2.833033928$ |
2.087179461 |
\( -\frac{1633157120}{14348907} a - \frac{2987130880}{4782969} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -721 a + 5541\) , \( -99160 a + 759442\bigr] \) |
${y}^2+{y}={x}^3-w{x}^2+\left(-721w+5541\right){x}-99160w+759442$ |
| 27.1-a3 |
27.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{26} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$0.527418005$ |
$14.16516964$ |
2.087179461 |
\( \frac{169467576320}{243} a - \frac{1297937039360}{243} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 3522 a - 26976\) , \( -304430 a + 2331604\bigr] \) |
${y}^2+{y}={x}^3+\left(3522w-26976\right){x}-304430w+2331604$ |
| 27.1-a4 |
27.1-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{22} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1.582254015$ |
$4.721723214$ |
2.087179461 |
\( \frac{1633157120}{14348907} a - \frac{10594549760}{14348907} \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( 39 a - 279\) , \( -430 a + 3276\bigr] \) |
${y}^2+{y}={x}^3-w{x}^2+\left(39w-279\right){x}-430w+3276$ |
| 27.1-b1 |
27.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.944344642$ |
1.319117816 |
\( -\frac{169467576320}{243} a - \frac{376156487680}{81} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 19 a - 129\) , \( 394 a - 3011\bigr] \) |
${y}^2+{y}={x}^3+w{x}^2+\left(19w-129\right){x}+394w-3011$ |
| 27.1-b2 |
27.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{34} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$1$ |
$2.833033928$ |
1.319117816 |
\( -\frac{1633157120}{14348907} a - \frac{2987130880}{4782969} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -81 a - 519\) , \( -1762 a - 11739\bigr] \) |
${y}^2+{y}={x}^3+w{x}^2+\left(-81w-519\right){x}-1762w-11739$ |
| 27.1-b3 |
27.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{26} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$14.16516964$ |
1.319117816 |
\( \frac{169467576320}{243} a - \frac{1297937039360}{243} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -168 a - 1146\) , \( 11732 a + 78177\bigr] \) |
${y}^2+{y}={x}^3+\left(-168w-1146\right){x}+11732w+78177$ |
| 27.1-b4 |
27.1-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{22} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.721723214$ |
1.319117816 |
\( \frac{1633157120}{14348907} a - \frac{10594549760}{14348907} \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( 19 a + 141\) , \( -356 a - 2381\bigr] \) |
${y}^2+{y}={x}^3+w{x}^2+\left(19w+141\right){x}-356w-2381$ |
| 27.2-a1 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{26} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$0.527418005$ |
$14.16516964$ |
2.087179461 |
\( -\frac{169467576320}{243} a - \frac{376156487680}{81} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 31 a - 340\) , \( -1400 a + 10962\bigr] \) |
${y}^2+{y}={x}^3+\left(w-1\right){x}^2+\left(31w-340\right){x}-1400w+10962$ |
| 27.2-a2 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{22} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$1.582254015$ |
$4.721723214$ |
2.087179461 |
\( -\frac{1633157120}{14348907} a - \frac{2987130880}{4782969} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -39 a - 240\) , \( 430 a + 2846\bigr] \) |
${y}^2+{y}={x}^3+\left(w-1\right){x}^2+\left(-39w-240\right){x}+430w+2846$ |
| 27.2-a3 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$7.911270077$ |
$0.944344642$ |
2.087179461 |
\( \frac{169467576320}{243} a - \frac{1297937039360}{243} \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( 391 a - 2980\) , \( 10278 a - 78708\bigr] \) |
${y}^2+{y}={x}^3+\left(w-1\right){x}^2+\left(391w-2980\right){x}+10278w-78708$ |
| 27.2-a4 |
27.2-a |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{34} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B, 5B |
$1$ |
\( 2 \) |
$2.637090025$ |
$2.833033928$ |
2.087179461 |
\( \frac{1633157120}{14348907} a - \frac{10594549760}{14348907} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 348 a - 2664\) , \( 13874 a - 106258\bigr] \) |
${y}^2+{y}={x}^3+\left(348w-2664\right){x}+13874w-106258$ |
| 27.2-b1 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{26} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$14.16516964$ |
1.319117816 |
\( -\frac{169467576320}{243} a - \frac{376156487680}{81} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -15139 a - 100790\) , \( 2746408 a + 18288101\bigr] \) |
${y}^2+{y}={x}^3+\left(-w+1\right){x}^2+\left(-15139w-100790\right){x}+2746408w+18288101$ |
| 27.2-b2 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{22} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.721723214$ |
1.319117816 |
\( -\frac{1633157120}{14348907} a - \frac{2987130880}{4782969} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -19 a + 160\) , \( 356 a - 2737\bigr] \) |
${y}^2+{y}={x}^3+\left(-w+1\right){x}^2+\left(-19w+160\right){x}+356w-2737$ |
| 27.2-b3 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{14} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.2, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.944344642$ |
1.319117816 |
\( \frac{169467576320}{243} a - \frac{1297937039360}{243} \) |
\( \bigl[0\) , \( -a + 1\) , \( 1\) , \( -19 a - 110\) , \( -394 a - 2617\bigr] \) |
${y}^2+{y}={x}^3+\left(-w+1\right){x}^2+\left(-19w-110\right){x}-394w-2617$ |
| 27.2-b4 |
27.2-b |
$4$ |
$15$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
27.2 |
\( 3^{3} \) |
\( 3^{34} \) |
$2.91646$ |
$(3,a), (3,a+2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3, 5$ |
3B.1.1, 5B |
$1$ |
\( 2^{2} \cdot 3 \cdot 5 \) |
$1$ |
$2.833033928$ |
1.319117816 |
\( \frac{1633157120}{14348907} a - \frac{10594549760}{14348907} \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( 168 a + 1116\) , \( 10948 a + 72899\bigr] \) |
${y}^2+{y}={x}^3+\left(168w+1116\right){x}+10948w+72899$ |
| 28.1-a1 |
28.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{16} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$14.36911988$ |
$0.985803285$ |
3.957341143 |
\( \frac{106286015625663357}{66465861139202} a - \frac{353222055694778922}{33232930569601} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 2724 a - 20863\) , \( 214622 a - 1643771\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(2724w-20863\right){x}+214622w-1643771$ |
| 28.1-a2 |
28.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{8} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$7.184559941$ |
$3.943213143$ |
3.957341143 |
\( -\frac{780348831047775}{23059204} a + \frac{5976664389642267}{23059204} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 2754 a - 21093\) , \( 209802 a - 1606855\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(2754w-21093\right){x}+209802w-1606855$ |
| 28.1-a3 |
28.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$3.592279970$ |
$0.985803285$ |
3.957341143 |
\( -\frac{78739019535686561757}{4802} a + \frac{301527552986672683194}{2401} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 44064 a - 337483\) , \( 13460390 a - 103091923\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(44064w-337483\right){x}+13460390w-103091923$ |
| 28.1-a4 |
28.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.592279970$ |
$15.77285257$ |
3.957341143 |
\( \frac{1785885975}{38416} a + \frac{3191649507}{9604} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( 174 a - 1333\) , \( 3170 a - 24279\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(174w-1333\right){x}+3170w-24279$ |
| 28.1-b1 |
28.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{16} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$5.458743565$ |
$0.985803285$ |
6.013481878 |
\( \frac{106286015625663357}{66465861139202} a - \frac{353222055694778922}{33232930569601} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( 30 a + 179\) , \( 204 a + 1275\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3-{x}^2+\left(30w+179\right){x}+204w+1275$ |
| 28.1-b2 |
28.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{8} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.458743565$ |
$3.943213143$ |
6.013481878 |
\( -\frac{780348831047775}{23059204} a + \frac{5976664389642267}{23059204} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -21\) , \( -12 a - 163\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3-{x}^2-21{x}-12w-163$ |
| 28.1-b3 |
28.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$21.83497426$ |
$0.985803285$ |
6.013481878 |
\( -\frac{78739019535686561757}{4802} a + \frac{301527552986672683194}{2401} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -30 a - 541\) , \( -756 a - 8257\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3-{x}^2+\left(-30w-541\right){x}-756w-8257$ |
| 28.1-b4 |
28.1-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.1 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+1), (2)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.364685891$ |
$15.77285257$ |
6.013481878 |
\( \frac{1785885975}{38416} a + \frac{3191649507}{9604} \) |
\( \bigl[a + 1\) , \( -1\) , \( a + 1\) , \( -1\) , \( 1\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+\left(w+1\right){y}={x}^3-{x}^2-{x}+1$ |
| 28.2-a1 |
28.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{16} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2 \) |
$14.36911988$ |
$0.985803285$ |
3.957341143 |
\( -\frac{106286015625663357}{66465861139202} a - \frac{600158095763894487}{66465861139202} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -2724 a - 18139\) , \( -214622 a - 1429149\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-2724w-18139\right){x}-214622w-1429149$ |
| 28.2-a2 |
28.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$3.592279970$ |
$15.77285257$ |
3.957341143 |
\( -\frac{1785885975}{38416} a + \frac{14552484003}{38416} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -174 a - 1159\) , \( -3170 a - 21109\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-174w-1159\right){x}-3170w-21109$ |
| 28.2-a3 |
28.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{8} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$7.184559941$ |
$3.943213143$ |
3.957341143 |
\( \frac{780348831047775}{23059204} a + \frac{1299078889648623}{5764801} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -2754 a - 18339\) , \( -209802 a - 1397053\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-2754w-18339\right){x}-209802w-1397053$ |
| 28.2-a4 |
28.2-a |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$3.592279970$ |
$0.985803285$ |
3.957341143 |
\( \frac{78739019535686561757}{4802} a + \frac{524316086437658804631}{4802} \) |
\( \bigl[1\) , \( -1\) , \( 1\) , \( -44064 a - 293419\) , \( -13460390 a - 89631533\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-{x}^2+\left(-44064w-293419\right){x}-13460390w-89631533$ |
| 28.2-b1 |
28.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{16} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$5.458743565$ |
$0.985803285$ |
6.013481878 |
\( -\frac{106286015625663357}{66465861139202} a - \frac{600158095763894487}{66465861139202} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -32 a + 211\) , \( -205 a + 1480\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3-w{x}^2+\left(-32w+211\right){x}-205w+1480$ |
| 28.2-b2 |
28.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{8} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$2$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1.364685891$ |
$15.77285257$ |
6.013481878 |
\( -\frac{1785885975}{38416} a + \frac{14552484003}{38416} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -2 a + 1\) , \( -a + 2\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3-w{x}^2+\left(-2w+1\right){x}-w+2$ |
| 28.2-b3 |
28.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{4} \cdot 7^{8} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$5.458743565$ |
$3.943213143$ |
6.013481878 |
\( \frac{780348831047775}{23059204} a + \frac{1299078889648623}{5764801} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -2 a - 19\) , \( 11 a - 174\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3-w{x}^2+\left(-2w-19\right){x}+11w-174$ |
| 28.2-b4 |
28.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
28.2 |
\( 2^{2} \cdot 7 \) |
\( 2^{2} \cdot 7^{4} \) |
$2.94310$ |
$(7,a+5), (2)$ |
$2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$21.83497426$ |
$0.985803285$ |
6.013481878 |
\( \frac{78739019535686561757}{4802} a + \frac{524316086437658804631}{4802} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( 28 a - 569\) , \( 755 a - 9012\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3-w{x}^2+\left(28w-569\right){x}+755w-9012$ |
| 36.1-a1 |
36.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{2} \cdot 3^{16} \) |
$3.13394$ |
$(3,a), (3,a+2), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$25.41455941$ |
1.775029824 |
\( \frac{18541}{54} a + \frac{21025}{9} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -8 a + 198\) , \( -680 a + 5605\bigr] \) |
${y}^2+w{x}{y}+w{y}={x}^3+w{x}^2+\left(-8w+198\right){x}-680w+5605$ |
| 36.1-b1 |
36.1-b |
$4$ |
$10$ |
\(\Q(\sqrt{205}) \) |
$2$ |
$[2, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{2} \) |
$3.13394$ |
$(3,a), (3,a+2), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2B, 5B.4.2 |
$1$ |
\( 2 \) |
$6.251824599$ |
$6.999515039$ |
3.056312836 |
\( -\frac{24389}{12} \) |
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 17 a + 69\) , \( 53 a + 261\bigr] \) |
${y}^2+\left(w+1\right){x}{y}+w{y}={x}^3+w{x}^2+\left(17w+69\right){x}+53w+261$ |
*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.
