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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1078.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1078.k1 | 1078g2 | \([1, 0, 0, -278223, -56508775]\) | \(413160293352625/42592\) | \(245534404192\) | \([]\) | \(5040\) | \(1.6153\) | |
1078.k2 | 1078g1 | \([1, 0, 0, -3823, -59207]\) | \(1071912625/360448\) | \(2077910990848\) | \([3]\) | \(1680\) | \(1.0660\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1078.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1078.k do not have complex multiplication.Modular form 1078.2.a.k
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.