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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1078.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1078.e1 | 1078a2 | \([1, 0, 1, -271, -1718]\) | \(911871625/10648\) | \(25565848\) | \([]\) | \(288\) | \(0.23367\) | |
1078.e2 | 1078a1 | \([1, 0, 1, -26, 46]\) | \(765625/22\) | \(52822\) | \([3]\) | \(96\) | \(-0.31563\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1078.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1078.e do not have complex multiplication.Modular form 1078.2.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.