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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 31680ci
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.u3 | 31680ci1 | \([0, 0, 0, -802668, -276790192]\) | \(299270638153369/1069200\) | \(204327301939200\) | \([2]\) | \(245760\) | \(1.9639\) | \(\Gamma_0(N)\)-optimal |
| 31680.u2 | 31680ci2 | \([0, 0, 0, -814188, -268435888]\) | \(312341975961049/17862322500\) | \(3413542988021760000\) | \([2, 2]\) | \(491520\) | \(2.3104\) | |
| 31680.u4 | 31680ci3 | \([0, 0, 0, 585492, -1095366832]\) | \(116149984977671/2779502343750\) | \(-531171169689600000000\) | \([2]\) | \(983040\) | \(2.6570\) | |
| 31680.u1 | 31680ci4 | \([0, 0, 0, -2398188, 1093170512]\) | \(7981893677157049/1917731420550\) | \(366484181635812556800\) | \([2]\) | \(983040\) | \(2.6570\) |
Rank
sage: E.rank()
The elliptic curves in class 31680ci have rank \(1\).
Complex multiplication
The elliptic curves in class 31680ci do not have complex multiplication.Modular form 31680.2.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 Cremona 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 Cremona labels.
