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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 30960cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
30960.bc3 | 30960cd1 | \([0, 0, 0, -9867, -376774]\) | \(35578826569/51600\) | \(154076774400\) | \([2]\) | \(36864\) | \(1.0488\) | \(\Gamma_0(N)\)-optimal |
30960.bc2 | 30960cd2 | \([0, 0, 0, -12747, -138886]\) | \(76711450249/41602500\) | \(124224399360000\) | \([2, 2]\) | \(73728\) | \(1.3954\) | |
30960.bc4 | 30960cd3 | \([0, 0, 0, 49173, -1092454]\) | \(4403686064471/2721093750\) | \(-8125142400000000\) | \([2]\) | \(147456\) | \(1.7419\) | |
30960.bc1 | 30960cd4 | \([0, 0, 0, -120747, 16039514]\) | \(65202655558249/512820150\) | \(1531272762777600\) | \([4]\) | \(147456\) | \(1.7419\) |
Rank
sage: E.rank()
The elliptic curves in class 30960cd have rank \(1\).
Complex multiplication
The elliptic curves in class 30960cd do not have complex multiplication.Modular form 30960.2.a.cd
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.