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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 714.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
714.i1 | 714i3 | \([1, 0, 0, -381702, -90803346]\) | \(-6150311179917589675873/244053849830826\) | \(-244053849830826\) | \([]\) | \(9720\) | \(1.8441\) | |
714.i2 | 714i2 | \([1, 0, 0, -972, -315144]\) | \(-101566487155393/42823570577256\) | \(-42823570577256\) | \([3]\) | \(3240\) | \(1.2948\) | |
714.i3 | 714i1 | \([1, 0, 0, 108, 11664]\) | \(139233463487/58763045376\) | \(-58763045376\) | \([9]\) | \(1080\) | \(0.74550\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 714.i have rank \(0\).
Complex multiplication
The elliptic curves in class 714.i do not have complex multiplication.Modular form 714.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.