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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 9075.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9075.a1 | 9075j1 | \([0, -1, 1, -1008, 12968]\) | \(-102400/3\) | \(-3321676875\) | \([]\) | \(8400\) | \(0.60585\) | \(\Gamma_0(N)\)-optimal |
9075.a2 | 9075j2 | \([0, -1, 1, 5042, -610182]\) | \(20480/243\) | \(-168159891796875\) | \([]\) | \(42000\) | \(1.4106\) |
Rank
sage: E.rank()
The elliptic curves in class 9075.a have rank \(1\).
Complex multiplication
The elliptic curves in class 9075.a do not have complex multiplication.Modular form 9075.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.