| 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 |
| 44.1-a1 |
44.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{2} \cdot 11 \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1.495802233$ |
$9.195017716$ |
3.134811134 |
\( -\frac{990711}{22} a - \frac{1916487}{11} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -a - 2\) , \( -a - 6\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a-2\right){x}-a-6$ |
| 44.1-a2 |
44.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{10} \cdot 11^{5} \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$0.299160446$ |
$9.195017716$ |
3.134811134 |
\( \frac{1517290830441}{42592} a - \frac{7415690928795}{42592} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 9 a - 17\) , \( -14 a + 108\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(9a-17\right){x}-14a+108$ |
| 44.1-b1 |
44.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{10} \cdot 11^{5} \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 5 \) |
$0.299160446$ |
$9.195017716$ |
3.134811134 |
\( -\frac{1517290830441}{42592} a - \frac{2949200049177}{21296} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -10 a - 7\) , \( 13 a + 95\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-10a-7\right){x}+13a+95$ |
| 44.1-b2 |
44.1-b |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{2} \cdot 11 \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1.495802233$ |
$9.195017716$ |
3.134811134 |
\( \frac{990711}{22} a - \frac{4823685}{22} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -2\) , \( -6\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}-2{x}-6$ |
| 44.1-c1 |
44.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{4} \cdot 11^{4} \) |
$2.01952$ |
$(a-6), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$9.831379401$ |
2.240779327 |
\( \frac{704969}{484} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( -4 a + 5\) , \( -3 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-4a+5\right){x}-3a+7$ |
| 44.1-c2 |
44.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{2} \cdot 11^{8} \) |
$2.01952$ |
$(a-6), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$9.831379401$ |
2.240779327 |
\( \frac{59776471}{29282} \) |
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 6 a - 45\) , \( 23 a - 121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(6a-45\right){x}+23a-121$ |
| 44.1-d1 |
44.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{10} \cdot 11^{5} \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 5 \) |
$2.037567229$ |
$1.211572561$ |
2.813299791 |
\( -\frac{1517290830441}{42592} a - \frac{2949200049177}{21296} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -270 a + 1307\) , \( 3859 a - 18867\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-270a+1307\right){x}+3859a-18867$ |
| 44.1-d2 |
44.1-d |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{2} \cdot 11 \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$10.18783614$ |
$30.28931404$ |
2.813299791 |
\( \frac{990711}{22} a - \frac{4823685}{22} \) |
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 5 a + 24\) , \( 7 a + 28\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(5a+24\right){x}+7a+28$ |
| 44.1-e1 |
44.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{4} \cdot 11^{4} \) |
$2.01952$ |
$(a-6), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$9.831379401$ |
2.240779327 |
\( \frac{704969}{484} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( 2 a + 3\) , \( 2 a + 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(2a+3\right){x}+2a+5$ |
| 44.1-e2 |
44.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{2} \cdot 11^{8} \) |
$2.01952$ |
$(a-6), (2)$ |
$0$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$9.831379401$ |
2.240779327 |
\( \frac{59776471}{29282} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -8 a - 37\) , \( -24 a - 97\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-8a-37\right){x}-24a-97$ |
| 44.1-f1 |
44.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{2} \cdot 11 \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$10.18783614$ |
$30.28931404$ |
2.813299791 |
\( -\frac{990711}{22} a - \frac{1916487}{11} \) |
\( \bigl[a\) , \( -a - 1\) , \( a + 1\) , \( -5 a + 10\) , \( -3 a + 6\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-5a+10\right){x}-3a+6$ |
| 44.1-f2 |
44.1-f |
$2$ |
$5$ |
\(\Q(\sqrt{77}) \) |
$2$ |
$[2, 0]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{10} \cdot 11^{5} \) |
$2.01952$ |
$(a-6), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 5 \) |
$2.037567229$ |
$1.211572561$ |
2.813299791 |
\( \frac{1517290830441}{42592} a - \frac{7415690928795}{42592} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 271 a + 1056\) , \( -3589 a - 13952\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(271a+1056\right){x}-3589a-13952$ |
*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.
