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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 137904e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137904.bx2 | 137904e1 | \([0, 1, 0, -26589, -1523898]\) | \(26919436288/2738853\) | \(211518724961232\) | \([2]\) | \(580608\) | \(1.4846\) | \(\Gamma_0(N)\)-optimal |
137904.bx1 | 137904e2 | \([0, 1, 0, -414444, -102831624]\) | \(6371214852688/77571\) | \(95851622640384\) | \([2]\) | \(1161216\) | \(1.8312\) |
Rank
sage: E.rank()
The elliptic curves in class 137904e have rank \(0\).
Complex multiplication
The elliptic curves in class 137904e do not have complex multiplication.Modular form 137904.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.