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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 33810.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.e1 | 33810h2 | \([1, 1, 0, -134488, 18927448]\) | \(5490138280852426201/2760\) | \(135240\) | \([]\) | \(95904\) | \(1.2203\) | |
33810.e2 | 33810h1 | \([1, 1, 0, -1663, 25243]\) | \(10389923853001/82127250\) | \(4024235250\) | \([]\) | \(31968\) | \(0.67097\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 33810.e have rank \(1\).
Complex multiplication
The elliptic curves in class 33810.e do not have complex multiplication.Modular form 33810.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.