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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 75.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75.a1 | 75c2 | \([0, 1, 1, -208, -1256]\) | \(-102400/3\) | \(-29296875\) | \([]\) | \(30\) | \(0.21162\) | |
75.a2 | 75c1 | \([0, 1, 1, 2, 4]\) | \(20480/243\) | \(-6075\) | \([5]\) | \(6\) | \(-0.59310\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75.a have rank \(0\).
Complex multiplication
The elliptic curves in class 75.a do not have complex multiplication.Modular form 75.2.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.