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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 2730c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2730.b4 | 2730c1 | \([1, 1, 0, -720043, 118835197]\) | \(41285728533151645510969/17760741842188800000\) | \(17760741842188800000\) | \([2]\) | \(115200\) | \(2.3900\) | \(\Gamma_0(N)\)-optimal |
| 2730.b2 | 2730c2 | \([1, 1, 0, -9859563, 11906988093]\) | \(105997782562506306791694649/51649016225625000000\) | \(51649016225625000000\) | \([2, 2]\) | \(230400\) | \(2.7366\) | |
| 2730.b1 | 2730c3 | \([1, 1, 0, -157734563, 762431763093]\) | \(434014578033107719741685694649/103121648659575000\) | \(103121648659575000\) | \([2]\) | \(460800\) | \(3.0831\) | |
| 2730.b3 | 2730c4 | \([1, 1, 0, -8216883, 16006788837]\) | \(-61354313914516350666047929/75227254486083984375000\) | \(-75227254486083984375000\) | \([2]\) | \(460800\) | \(3.0831\) |
Rank
sage: E.rank()
The elliptic curves in class 2730c have rank \(0\).
Complex multiplication
The elliptic curves in class 2730c do not have complex multiplication.Modular form 2730.2.a.c
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
