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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 405600ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405600.ca3 | 405600ca1 | \([0, -1, 0, -280258, -53163488]\) | \(504358336/38025\) | \(183539412225000000\) | \([2, 2]\) | \(3096576\) | \(2.0578\) | \(\Gamma_0(N)\)-optimal* |
405600.ca2 | 405600ca2 | \([0, -1, 0, -914008, 273851512]\) | \(2186875592/428415\) | \(16543019021880000000\) | \([2]\) | \(6193152\) | \(2.4044\) | \(\Gamma_0(N)\)-optimal* |
405600.ca4 | 405600ca3 | \([0, -1, 0, 268992, -236612988]\) | \(55742968/658125\) | \(-25413149385000000000\) | \([2]\) | \(6193152\) | \(2.4044\) | |
405600.ca1 | 405600ca4 | \([0, -1, 0, -4399633, -3550512863]\) | \(30488290624/195\) | \(60238576320000000\) | \([2]\) | \(6193152\) | \(2.4044\) |
Rank
sage: E.rank()
The elliptic curves in class 405600ca have rank \(0\).
Complex multiplication
The elliptic curves in class 405600ca do not have complex multiplication.Modular form 405600.2.a.ca
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.