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SageMath
E = EllipticCurve("fr1")
E.isogeny_class()
Elliptic curves in class 103488fr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
103488.j1 | 103488fr1 | \([0, -1, 0, -737, 11649]\) | \(-3451273/2376\) | \(-30519853056\) | \([]\) | \(82944\) | \(0.71213\) | \(\Gamma_0(N)\)-optimal |
103488.j2 | 103488fr2 | \([0, -1, 0, 5983, -172479]\) | \(1843623047/2044416\) | \(-26260638007296\) | \([]\) | \(248832\) | \(1.2614\) |
Rank
sage: E.rank()
The elliptic curves in class 103488fr have rank \(1\).
Complex multiplication
The elliptic curves in class 103488fr do not have complex multiplication.Modular form 103488.2.a.fr
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.