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SageMath
E = EllipticCurve("ec1")
E.isogeny_class()
Elliptic curves in class 73920ec
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.l3 | 73920ec1 | \([0, -1, 0, -1039521, 408288321]\) | \(473897054735271721/779625\) | \(204374016000\) | \([2]\) | \(589824\) | \(1.8621\) | \(\Gamma_0(N)\)-optimal |
| 73920.l2 | 73920ec2 | \([0, -1, 0, -1039841, 408024705]\) | \(474334834335054841/607815140625\) | \(159335092224000000\) | \([2, 2]\) | \(1179648\) | \(2.2086\) | |
| 73920.l4 | 73920ec3 | \([0, -1, 0, -759841, 632416705]\) | \(-185077034913624841/551466161890875\) | \(-144563545542721536000\) | \([2]\) | \(2359296\) | \(2.5552\) | |
| 73920.l1 | 73920ec4 | \([0, -1, 0, -1324961, 166756161]\) | \(981281029968144361/522287841796875\) | \(136914624000000000000\) | \([2]\) | \(2359296\) | \(2.5552\) |
Rank
sage: E.rank()
The elliptic curves in class 73920ec have rank \(0\).
Complex multiplication
The elliptic curves in class 73920ec do not have complex multiplication.Modular form 73920.2.a.ec
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.
