| 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 |
| 170.1-a1 |
170.1-a |
$8$ |
$16$ |
3.3.148.1 |
$3$ |
$[3, 0]$ |
170.1 |
\( 2 \cdot 5 \cdot 17 \) |
\( 2^{16} \cdot 5^{4} \cdot 17 \) |
$2.55865$ |
$(a^2-a-2), (a^2-a-1), (2a+1)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$102.1673881$ |
2.099526893 |
\( -\frac{1879071913}{680000} a^{2} - \frac{554741277}{85000} a + \frac{1405376861}{680000} \) |
\( \bigl[a^{2} - 2\) , \( a - 1\) , \( a + 1\) , \( -18762010943285 a^{2} - 21953175466668 a + 8645743411111\) , \( 84251758311077222632 a^{2} + 98581843874274084204 a - 38824147714992035994\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-18762010943285a^{2}-21953175466668a+8645743411111\right){x}+84251758311077222632a^{2}+98581843874274084204a-38824147714992035994$ |
| 2.2-a2 |
2.2-a |
$8$ |
$16$ |
3.3.316.1 |
$3$ |
$[3, 0]$ |
2.2 |
\( 2 \) |
\( 2^{16} \) |
$1.78301$ |
$(-a+1)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$171.4772930$ |
0.602896961 |
\( \frac{4108077233}{65536} a^{2} - \frac{5810733523}{32768} a + \frac{2294990397}{32768} \) |
\( \bigl[a^{2} + a - 3\) , \( -a\) , \( a\) , \( -608022265892705467026233 a^{2} + 321836266424514724186808 a + 2583572058090078441238362\) , \( -627730820434767876188418741768082419 a^{2} + 332268331114001902947589285516929027 a + 2667316476142693080354081387519995775\bigr] \) |
${y}^2+\left(a^{2}+a-3\right){x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-608022265892705467026233a^{2}+321836266424514724186808a+2583572058090078441238362\right){x}-627730820434767876188418741768082419a^{2}+332268331114001902947589285516929027a+2667316476142693080354081387519995775$ |
| 30.1-i8 |
30.1-i |
$8$ |
$16$ |
3.3.837.1 |
$3$ |
$[3, 0]$ |
30.1 |
\( 2 \cdot 3 \cdot 5 \) |
\( 2^{16} \cdot 3^{16} \cdot 5^{4} \) |
$4.55710$ |
$(-a^2+a+4), (a+2), (a-2)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{10} \) |
$1$ |
$15.96797761$ |
2.207736193 |
\( \frac{15826879527157}{29859840000} a^{2} - \frac{25363542673001}{29859840000} a - \frac{20860086325649}{29859840000} \) |
\( \bigl[1\) , \( 0\) , \( 0\) , \( 38420951338 a^{2} + 97163433775 a + 15192644438\) , \( -15547313570172292 a^{2} - 39317880475295023 a - 6147812555693206\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(38420951338a^{2}+97163433775a+15192644438\right){x}-15547313570172292a^{2}-39317880475295023a-6147812555693206$ |
| 45.1-b6 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$985.1451572$ |
0.917854808 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 45.1-b9 |
45.1-b |
$10$ |
$32$ |
\(\Q(\zeta_{15})^+\) |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$4.82355$ |
$(-a-1), (-a^3+a^2+3a-2)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$985.1451572$ |
0.917854808 |
\( -\frac{1}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}$ |
| 405.1-c6 |
405.1-c |
$10$ |
$32$ |
\(\Q(\zeta_{20})^+\) |
$4$ |
$[4, 0]$ |
405.1 |
\( 3^{4} \cdot 5 \) |
\( 3^{4} \cdot 5^{4} \) |
$8.46419$ |
$(a), (3)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{2} \) |
$1$ |
$985.1451572$ |
1.376782212 |
\( \frac{56667352321}{15} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -80\) , \( 242\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-80{x}+242$ |
| 10.1-c2 |
10.1-c |
$8$ |
$16$ |
4.4.6809.1 |
$4$ |
$[4, 0]$ |
10.1 |
\( 2 \cdot 5 \) |
\( - 2^{16} \cdot 5^{4} \) |
$9.83287$ |
$(a+1), (a^3-4a)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$704.2378368$ |
2.133623059 |
\( \frac{4752624697430737}{40960000} a^{3} - \frac{2765657876476891}{40960000} a^{2} - \frac{5539826641613443}{10240000} a + \frac{8158468803972659}{40960000} \) |
\( \bigl[a^{2} - 2\) , \( -a^{3} - a^{2} + 5 a + 4\) , \( a\) , \( -15 a^{3} + 4 a^{2} + 71 a - 9\) , \( 20 a^{3} - 14 a^{2} - 95 a + 50\bigr] \) |
${y}^2+\left(a^{2}-2\right){x}{y}+a{y}={x}^{3}+\left(-a^{3}-a^{2}+5a+4\right){x}^{2}+\left(-15a^{3}+4a^{2}+71a-9\right){x}+20a^{3}-14a^{2}-95a+50$ |
| 24.1-e2 |
24.1-e |
$8$ |
$16$ |
4.4.13824.1 |
$4$ |
$[4, 0]$ |
24.1 |
\( 2^{3} \cdot 3 \) |
\( 2^{8} \cdot 3^{8} \) |
$15.63082$ |
$(a-2), (a^2+a-3)$ |
0 |
$\Z/16\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1033.672610$ |
1.098945510 |
\( \frac{3065617154}{9} \) |
\( \bigl[a^{3} + a^{2} - 3 a - 2\) , \( -a^{3} + a^{2} + 4 a - 3\) , \( a^{2} - 2\) , \( 963 a^{3} - 1966 a^{2} - 1350 a + 2354\) , \( -41085 a^{3} + 88721 a^{2} + 52790 a - 111712\bigr] \) |
${y}^2+\left(a^{3}+a^{2}-3a-2\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(963a^{3}-1966a^{2}-1350a+2354\right){x}-41085a^{3}+88721a^{2}+52790a-111712$ |
*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.
