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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 80850.ci
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80850.ci1 | 80850by4 | \([1, 0, 1, -60368026, 180528788948]\) | \(13235378341603461121/9240\) | \(16985574375000\) | \([2]\) | \(3538944\) | \(2.7516\) | |
80850.ci2 | 80850by2 | \([1, 0, 1, -3773026, 2820488948]\) | \(3231355012744321/85377600\) | \(156946707225000000\) | \([2, 2]\) | \(1769472\) | \(2.4051\) | |
80850.ci3 | 80850by3 | \([1, 0, 1, -3626026, 3050396948]\) | \(-2868190647517441/527295615000\) | \(-969309403267734375000\) | \([2]\) | \(3538944\) | \(2.7516\) | |
80850.ci4 | 80850by1 | \([1, 0, 1, -245026, 40424948]\) | \(885012508801/127733760\) | \(234808580160000000\) | \([2]\) | \(884736\) | \(2.0585\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 80850.ci have rank \(1\).
Complex multiplication
The elliptic curves in class 80850.ci do not have complex multiplication.Modular form 80850.2.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.