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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 317900e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 317900.e4 | 317900e1 | \([0, 1, 0, -327533, 70560688]\) | \(643956736/15125\) | \(91270182781250000\) | \([2]\) | \(4478976\) | \(2.0399\) | \(\Gamma_0(N)\)-optimal |
| 317900.e3 | 317900e2 | \([0, 1, 0, -724908, -134484812]\) | \(436334416/171875\) | \(16594578687500000000\) | \([2]\) | \(8957952\) | \(2.3865\) | |
| 317900.e2 | 317900e3 | \([0, 1, 0, -3217533, -2194476812]\) | \(610462990336/8857805\) | \(53451469844011250000\) | \([2]\) | \(13436928\) | \(2.5892\) | |
| 317900.e1 | 317900e4 | \([0, 1, 0, -51299908, -141441034812]\) | \(154639330142416/33275\) | \(3212710433900000000\) | \([2]\) | \(26873856\) | \(2.9358\) |
Rank
sage: E.rank()
The elliptic curves in class 317900e have rank \(1\).
Complex multiplication
The elliptic curves in class 317900e do not have complex multiplication.Modular form 317900.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
