Show commands:
SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 67830.s
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67830.s1 | 67830p8 | \([1, 0, 1, -2903932529, 60196943047556]\) | \(2708215857449597952771459256806409/1815677562935478375000000000\) | \(1815677562935478375000000000\) | \([2]\) | \(59719680\) | \(4.1690\) | |
| 67830.s2 | 67830p5 | \([1, 0, 1, -2903465114, 60217301097812]\) | \(2706908330196708836642873424493849/816939805815000\) | \(816939805815000\) | \([6]\) | \(19906560\) | \(3.6197\) | |
| 67830.s3 | 67830p6 | \([1, 0, 1, -217399409, 541937811332]\) | \(1136315122909965387044499819529/530704359775758422016000000\) | \(530704359775758422016000000\) | \([2, 2]\) | \(29859840\) | \(3.8224\) | |
| 67830.s4 | 67830p2 | \([1, 0, 1, -181466594, 940883727476]\) | \(660866552951225193140994678169/363054521201227329600\) | \(363054521201227329600\) | \([2, 6]\) | \(9953280\) | \(3.2731\) | |
| 67830.s5 | 67830p4 | \([1, 0, 1, -180445994, 951990304916]\) | \(-649778658927959232413187423769/15498405515425377751317720\) | \(-15498405515425377751317720\) | \([6]\) | \(19906560\) | \(3.6197\) | |
| 67830.s6 | 67830p3 | \([1, 0, 1, -111231089, -445724835964]\) | \(152195662006675487969752714249/2254051004206282702848000\) | \(2254051004206282702848000\) | \([2]\) | \(14929920\) | \(3.4759\) | |
| 67830.s7 | 67830p1 | \([1, 0, 1, -11405474, 14526794612]\) | \(164083032511008797673646489/3779535863669623787520\) | \(3779535863669623787520\) | \([6]\) | \(4976640\) | \(2.9266\) | \(\Gamma_0(N)\)-optimal |
| 67830.s8 | 67830p7 | \([1, 0, 1, 770440591, 4097766675332]\) | \(50575615882668425252678113940471/36522079745400816582633408000\) | \(-36522079745400816582633408000\) | \([2]\) | \(59719680\) | \(4.1690\) |
Rank
sage: E.rank()
The elliptic curves in class 67830.s have rank \(1\).
Complex multiplication
The elliptic curves in class 67830.s do not have complex multiplication.Modular form 67830.2.a.s
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 6 & 12 & 4 & 12 & 4 \\ 3 & 1 & 6 & 2 & 4 & 12 & 4 & 12 \\ 2 & 6 & 1 & 3 & 6 & 2 & 6 & 2 \\ 6 & 2 & 3 & 1 & 2 & 6 & 2 & 6 \\ 12 & 4 & 6 & 2 & 1 & 12 & 4 & 3 \\ 4 & 12 & 2 & 6 & 12 & 1 & 3 & 4 \\ 12 & 4 & 6 & 2 & 4 & 3 & 1 & 12 \\ 4 & 12 & 2 & 6 & 3 & 4 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
