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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 87120cs
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
87120.w2 | 87120cs1 | \([0, 0, 0, -13620123, -9627306678]\) | \(1469878353/640000\) | \(121664921670655672320000\) | \([2]\) | \(6082560\) | \(3.1257\) | \(\Gamma_0(N)\)-optimal |
87120.w1 | 87120cs2 | \([0, 0, 0, -105618843, 411156438858]\) | \(685429074513/12500000\) | \(2376268001379993600000000\) | \([2]\) | \(12165120\) | \(3.4723\) |
Rank
sage: E.rank()
The elliptic curves in class 87120cs have rank \(0\).
Complex multiplication
The elliptic curves in class 87120cs do not have complex multiplication.Modular form 87120.2.a.cs
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.