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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 75.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75.c1 | 75a1 | \([0, -1, 1, -8, -7]\) | \(-102400/3\) | \(-1875\) | \([]\) | \(6\) | \(-0.59310\) | \(\Gamma_0(N)\)-optimal |
75.c2 | 75a2 | \([0, -1, 1, 42, 443]\) | \(20480/243\) | \(-94921875\) | \([]\) | \(30\) | \(0.21162\) |
Rank
sage: E.rank()
The elliptic curves in class 75.c have rank \(0\).
Complex multiplication
The elliptic curves in class 75.c do not have complex multiplication.Modular form 75.2.a.c
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.