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SageMath
E = EllipticCurve("ct1")
E.isogeny_class()
Elliptic curves in class 115920ct
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
115920.r2 | 115920ct1 | \([0, 0, 0, -232203, 43066298]\) | \(463702796512201/15214500\) | \(45430253568000\) | \([2]\) | \(552960\) | \(1.7143\) | \(\Gamma_0(N)\)-optimal |
115920.r3 | 115920ct2 | \([0, 0, 0, -222123, 46975322]\) | \(-405897921250921/84358968750\) | \(-251894530944000000\) | \([2]\) | \(1105920\) | \(2.0609\) | |
115920.r1 | 115920ct3 | \([0, 0, 0, -415803, -33939862]\) | \(2662558086295801/1374177967680\) | \(4103273424644997120\) | \([2]\) | \(1658880\) | \(2.2636\) | |
115920.r4 | 115920ct4 | \([0, 0, 0, 1559877, -263513878]\) | \(140574743422291079/91397357868600\) | \(-272911048237913702400\) | \([2]\) | \(3317760\) | \(2.6102\) |
Rank
sage: E.rank()
The elliptic curves in class 115920ct have rank \(1\).
Complex multiplication
The elliptic curves in class 115920ct do not have complex multiplication.Modular form 115920.2.a.ct
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.