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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 45325.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45325.k1 | 45325c3 | \([0, 1, 1, -2294833, 1337292744]\) | \(727057727488000/37\) | \(68015828125\) | \([]\) | \(326592\) | \(1.9998\) | |
| 45325.k2 | 45325c2 | \([0, 1, 1, -28583, 1791619]\) | \(1404928000/50653\) | \(93113668703125\) | \([]\) | \(108864\) | \(1.4504\) | |
| 45325.k3 | 45325c1 | \([0, 1, 1, -4083, -101006]\) | \(4096000/37\) | \(68015828125\) | \([]\) | \(36288\) | \(0.90114\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 45325.k have rank \(0\).
Complex multiplication
The elliptic curves in class 45325.k do not have complex multiplication.Modular form 45325.2.a.k
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
