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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 34225.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34225.e1 | 34225b3 | \([0, -1, 1, -64114833, -197578150557]\) | \(727057727488000/37\) | \(1483310580203125\) | \([]\) | \(1181952\) | \(2.8323\) | |
34225.e2 | 34225b2 | \([0, -1, 1, -798583, -265720682]\) | \(1404928000/50653\) | \(2030652184298078125\) | \([]\) | \(393984\) | \(2.2830\) | |
34225.e3 | 34225b1 | \([0, -1, 1, -114083, 14753193]\) | \(4096000/37\) | \(1483310580203125\) | \([]\) | \(131328\) | \(1.7336\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34225.e have rank \(1\).
Complex multiplication
The elliptic curves in class 34225.e do not have complex multiplication.Modular form 34225.2.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 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.