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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 1078.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1078.i1 | 1078j2 | \([1, 1, 1, -5678, 162315]\) | \(413160293352625/42592\) | \(2087008\) | \([]\) | \(720\) | \(0.64230\) | |
1078.i2 | 1078j1 | \([1, 1, 1, -78, 139]\) | \(1071912625/360448\) | \(17661952\) | \([]\) | \(240\) | \(0.092997\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1078.i have rank \(1\).
Complex multiplication
The elliptic curves in class 1078.i do not have complex multiplication.Modular form 1078.2.a.i
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.