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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1040.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1040.c1 | 1040b1 | \([0, 1, 0, -20, 28]\) | \(3631696/65\) | \(16640\) | \([2]\) | \(64\) | \(-0.39434\) | \(\Gamma_0(N)\)-optimal |
1040.c2 | 1040b2 | \([0, 1, 0, 0, 100]\) | \(-4/4225\) | \(-4326400\) | \([2]\) | \(128\) | \(-0.047762\) |
Rank
sage: E.rank()
The elliptic curves in class 1040.c have rank \(1\).
Complex multiplication
The elliptic curves in class 1040.c do not have complex multiplication.Modular form 1040.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.