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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 455175.ci
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
455175.ci1 | 455175ci4 | \([1, -1, 1, -66001730, -159249314478]\) | \(115650783909361/27072079335\) | \(7443251552346323317734375\) | \([2]\) | \(84934656\) | \(3.4843\) | |
455175.ci2 | 455175ci2 | \([1, -1, 1, -22109855, 37912988022]\) | \(4347507044161/258084225\) | \(70958192187469362890625\) | \([2, 2]\) | \(42467328\) | \(3.1377\) | |
455175.ci3 | 455175ci1 | \([1, -1, 1, -21784730, 39141310272]\) | \(4158523459441/16065\) | \(4416943180047890625\) | \([2]\) | \(21233664\) | \(2.7911\) | \(\Gamma_0(N)\)-optimal* |
455175.ci4 | 455175ci3 | \([1, -1, 1, 16580020, 156458765022]\) | \(1833318007919/39525924375\) | \(-10867336576610328896484375\) | \([2]\) | \(84934656\) | \(3.4843\) |
Rank
sage: E.rank()
The elliptic curves in class 455175.ci have rank \(1\).
Complex multiplication
The elliptic curves in class 455175.ci do not have complex multiplication.Modular form 455175.2.a.ci
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.