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SageMath
E = EllipticCurve("ge1")
E.isogeny_class()
Elliptic curves in class 73920ge
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 73920.ep5 | 73920ge1 | \([0, 1, 0, 9219, -1118925]\) | \(84611246065664/580054565475\) | \(-593975875046400\) | \([2]\) | \(262144\) | \(1.5165\) | \(\Gamma_0(N)\)-optimal |
| 73920.ep4 | 73920ge2 | \([0, 1, 0, -122001, -15002001]\) | \(12257375872392016/1191317675625\) | \(19518548797440000\) | \([2, 2]\) | \(524288\) | \(1.8631\) | |
| 73920.ep3 | 73920ge3 | \([0, 1, 0, -439521, 95431455]\) | \(143279368983686884/22699269140625\) | \(1487619302400000000\) | \([2, 2]\) | \(1048576\) | \(2.2096\) | |
| 73920.ep2 | 73920ge4 | \([0, 1, 0, -1904001, -1011852801]\) | \(11647843478225136004/128410942275\) | \(8415539512934400\) | \([2]\) | \(1048576\) | \(2.2096\) | |
| 73920.ep6 | 73920ge5 | \([0, 1, 0, 780159, 531832959]\) | \(400647648358480318/1163177490234375\) | \(-152460000000000000000\) | \([2]\) | \(2097152\) | \(2.5562\) | |
| 73920.ep1 | 73920ge6 | \([0, 1, 0, -6739521, 6731851455]\) | \(258286045443018193442/8440380939375\) | \(1106297610485760000\) | \([2]\) | \(2097152\) | \(2.5562\) |
Rank
sage: E.rank()
The elliptic curves in class 73920ge have rank \(1\).
Complex multiplication
The elliptic curves in class 73920ge do not have complex multiplication.Modular form 73920.2.a.ge
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
