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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 185020f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
185020.h2 | 185020f1 | \([0, -1, 0, -7965, -272150]\) | \(-4153551486976/20796875\) | \(-279842750000\) | \([]\) | \(233280\) | \(1.0438\) | \(\Gamma_0(N)\)-optimal |
185020.h1 | 185020f2 | \([0, -1, 0, -645965, -199615250]\) | \(-2215314389248638976/275\) | \(-3700400\) | \([]\) | \(699840\) | \(1.5931\) |
Rank
sage: E.rank()
The elliptic curves in class 185020f have rank \(1\).
Complex multiplication
The elliptic curves in class 185020f do not have complex multiplication.Modular form 185020.2.a.f
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.