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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 100014.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100014.e1 | 100014f2 | \([1, 0, 0, -87037, -9623389]\) | \(72918170522696196433/2250945132306174\) | \(2250945132306174\) | \([]\) | \(1213920\) | \(1.7214\) | |
100014.e2 | 100014f1 | \([1, 0, 0, -11707, 482009]\) | \(177444640175483953/1913455446264\) | \(1913455446264\) | \([3]\) | \(404640\) | \(1.1721\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 100014.e have rank \(1\).
Complex multiplication
The elliptic curves in class 100014.e do not have complex multiplication.Modular form 100014.2.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 LMFDB 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 LMFDB labels.