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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 18720.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
18720.e1 | 18720w2 | \([0, 0, 0, -348, 2272]\) | \(42144192/4225\) | \(467251200\) | \([2]\) | \(6144\) | \(0.39980\) | |
18720.e2 | 18720w1 | \([0, 0, 0, 27, 172]\) | \(1259712/8125\) | \(-14040000\) | \([2]\) | \(3072\) | \(0.053229\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 18720.e have rank \(2\).
Complex multiplication
The elliptic curves in class 18720.e do not have complex multiplication.Modular form 18720.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.