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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1225e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1225.f2 | 1225e1 | \([1, 1, 0, -9825, -412250]\) | \(-9317\) | \(-11259376953125\) | \([]\) | \(1680\) | \(1.2427\) | \(\Gamma_0(N)\)-optimal |
1225.f1 | 1225e2 | \([1, 1, 0, -254901700, 1566310159625]\) | \(-162677523113838677\) | \(-11259376953125\) | \([]\) | \(62160\) | \(3.0482\) |
Rank
sage: E.rank()
The elliptic curves in class 1225e have rank \(0\).
Complex multiplication
The elliptic curves in class 1225e do not have complex multiplication.Modular form 1225.2.a.e
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.
\(\left(\begin{array}{rr} 1 & 37 \\ 37 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.