| 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 |
| 6084.2-a1 |
6084.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 13^{4} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$0.294828824$ |
$3.590171887$ |
2.245388215 |
\( \frac{3631696}{507} \) |
\( \bigl[a\) , \( 1\) , \( 0\) , \( -5\) , \( 3\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+{x}^{2}-5{x}+3$ |
| 6084.2-a2 |
6084.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{4} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.147414412$ |
$3.590171887$ |
2.245388215 |
\( \frac{1048576}{117} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -5\) , \( 6\bigr] \) |
${y}^2={x}^{3}-{x}^{2}-5{x}+6$ |
| 6084.2-b1 |
6084.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{19} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.643404292$ |
1.819822151 |
\( \frac{253048096475372}{559607373} a - \frac{218119524144004}{559607373} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 73 a - 348\) , \( 646 a - 2293\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(73a-348\right){x}+646a-2293$ |
| 6084.2-b2 |
6084.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{7} \cdot 13^{8} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.643404292$ |
1.819822151 |
\( -\frac{141982147868}{2313441} a - \frac{41954607644}{2313441} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 143 a - 158\) , \( -1232 a + 293\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(143a-158\right){x}-1232a+293$ |
| 6084.2-b3 |
6084.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{14} \cdot 13^{4} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.286808584$ |
1.819822151 |
\( \frac{417240032}{1108809} a + \frac{1399915568}{1108809} \) |
\( \bigl[a\) , \( a - 1\) , \( 0\) , \( 8 a - 23\) , \( -17 a - 31\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(8a-23\right){x}-17a-31$ |
| 6084.2-b4 |
6084.2-b |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{16} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.286808584$ |
1.819822151 |
\( -\frac{3919376384}{6908733} a + \frac{14220271616}{6908733} \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -14 a + 22\) , \( 10 a + 49\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-14a+22\right){x}+10a+49$ |
| 6084.2-c1 |
6084.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{19} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.643404292$ |
1.819822151 |
\( -\frac{253048096475372}{559607373} a - \frac{218119524144004}{559607373} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -73 a - 348\) , \( -646 a - 2293\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-73a-348\right){x}-646a-2293$ |
| 6084.2-c2 |
6084.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{7} \cdot 13^{8} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$0.643404292$ |
1.819822151 |
\( \frac{141982147868}{2313441} a - \frac{41954607644}{2313441} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -143 a - 158\) , \( 1232 a + 293\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-143a-158\right){x}+1232a+293$ |
| 6084.2-c3 |
6084.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{14} \cdot 13^{4} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$1.286808584$ |
1.819822151 |
\( -\frac{417240032}{1108809} a + \frac{1399915568}{1108809} \) |
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( -8 a - 23\) , \( 17 a - 31\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-8a-23\right){x}+17a-31$ |
| 6084.2-c4 |
6084.2-c |
$4$ |
$4$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{16} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.286808584$ |
1.819822151 |
\( \frac{3919376384}{6908733} a + \frac{14220271616}{6908733} \) |
\( \bigl[0\) , \( -a\) , \( 0\) , \( 14 a + 22\) , \( -10 a + 49\bigr] \) |
${y}^2={x}^{3}-a{x}^{2}+\left(14a+22\right){x}-10a+49$ |
| 6084.2-d1 |
6084.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{16} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \cdot 7 \) |
$0.124019412$ |
$1.290412728$ |
4.752833511 |
\( \frac{34153369600}{62178597} a - \frac{43076829184}{62178597} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -10 a + 19\) , \( -40 a - 38\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-10a+19\right){x}-40a-38$ |
| 6084.2-d2 |
6084.2-d |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{8} \cdot 13^{4} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \cdot 7 \) |
$0.248038825$ |
$1.290412728$ |
4.752833511 |
\( -\frac{582894288160}{369603} a + \frac{400569131728}{369603} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( -40 a + 91\) , \( 209 a + 301\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(-40a+91\right){x}+209a+301$ |
| 6084.2-e1 |
6084.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{12} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \cdot 3^{3} \) |
$0.615690421$ |
$1.882955324$ |
4.918567838 |
\( \frac{16384000}{9477} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -13\) , \( -4\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-13{x}-4$ |
| 6084.2-e2 |
6084.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{2} \cdot 13^{12} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \cdot 3 \) |
$3.694142529$ |
$0.627651774$ |
4.918567838 |
\( \frac{181037698000}{14480427} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -186\) , \( 946\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-186{x}+946$ |
| 6084.2-e3 |
6084.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{6} \cdot 13^{4} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$1.231380843$ |
$1.882955324$ |
4.918567838 |
\( \frac{1409938000}{4563} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -36\) , \( -80\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-36{x}-80$ |
| 6084.2-e4 |
6084.2-e |
$4$ |
$6$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{4} \cdot 13^{6} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \cdot 3 \) |
$1.847071264$ |
$0.627651774$ |
4.918567838 |
\( \frac{2725888000000}{19773} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -733\) , \( -7888\bigr] \) |
${y}^2={x}^{3}+{x}^{2}-733{x}-7888$ |
| 6084.2-f1 |
6084.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 3^{16} \cdot 13^{2} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \cdot 7 \) |
$0.124019412$ |
$1.290412728$ |
4.752833511 |
\( -\frac{34153369600}{62178597} a - \frac{43076829184}{62178597} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 10 a + 19\) , \( 40 a - 38\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(10a+19\right){x}+40a-38$ |
| 6084.2-f2 |
6084.2-f |
$2$ |
$2$ |
\(\Q(\sqrt{-2}) \) |
$2$ |
$[0, 1]$ |
6084.2 |
\( 2^{2} \cdot 3^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 3^{8} \cdot 13^{4} \) |
$2.23219$ |
$(a), (-a-1), (a-1), (13)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \cdot 7 \) |
$0.248038825$ |
$1.290412728$ |
4.752833511 |
\( \frac{582894288160}{369603} a + \frac{400569131728}{369603} \) |
\( \bigl[a\) , \( 0\) , \( 0\) , \( 40 a + 91\) , \( -209 a + 301\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(40a+91\right){x}-209a+301$ |
*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.
