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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 8450e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8450.j1 | 8450e1 | \([1, 1, 0, -107825, 13789625]\) | \(-2941225/52\) | \(-2451113945312500\) | \([]\) | \(60480\) | \(1.7500\) | \(\Gamma_0(N)\)-optimal |
8450.j2 | 8450e2 | \([1, 1, 0, 420300, 66074000]\) | \(174196775/140608\) | \(-6627812108125000000\) | \([]\) | \(181440\) | \(2.2993\) |
Rank
sage: E.rank()
The elliptic curves in class 8450e have rank \(1\).
Complex multiplication
The elliptic curves in class 8450e do not have complex multiplication.Modular form 8450.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.