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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 2850.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2850.g1 | 2850e3 | \([1, 1, 0, -10700, -430500]\) | \(8671983378625/82308\) | \(1286062500\) | \([2]\) | \(5184\) | \(0.91022\) | |
2850.g2 | 2850e4 | \([1, 1, 0, -10450, -451250]\) | \(-8078253774625/846825858\) | \(-13231654031250\) | \([2]\) | \(10368\) | \(1.2568\) | |
2850.g3 | 2850e1 | \([1, 1, 0, -200, 0]\) | \(57066625/32832\) | \(513000000\) | \([2]\) | \(1728\) | \(0.36091\) | \(\Gamma_0(N)\)-optimal |
2850.g4 | 2850e2 | \([1, 1, 0, 800, 1000]\) | \(3616805375/2105352\) | \(-32896125000\) | \([2]\) | \(3456\) | \(0.70749\) |
Rank
sage: E.rank()
The elliptic curves in class 2850.g have rank \(0\).
Complex multiplication
The elliptic curves in class 2850.g do not have complex multiplication.Modular form 2850.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.