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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 193200ca
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193200.dx3 | 193200ca1 | \([0, 1, 0, -141408, 20419188]\) | \(4886171981209/270480\) | \(17310720000000\) | \([2]\) | \(884736\) | \(1.6057\) | \(\Gamma_0(N)\)-optimal |
193200.dx2 | 193200ca2 | \([0, 1, 0, -149408, 17971188]\) | \(5763259856089/1143116100\) | \(73159430400000000\) | \([2, 2]\) | \(1769472\) | \(1.9523\) | |
193200.dx4 | 193200ca3 | \([0, 1, 0, 310592, 107211188]\) | \(51774168853511/107398242630\) | \(-6873487528320000000\) | \([2]\) | \(3538944\) | \(2.2989\) | |
193200.dx1 | 193200ca4 | \([0, 1, 0, -737408, -227812812]\) | \(692895692874169/51420783750\) | \(3290930160000000000\) | \([2]\) | \(3538944\) | \(2.2989\) |
Rank
sage: E.rank()
The elliptic curves in class 193200ca have rank \(0\).
Complex multiplication
The elliptic curves in class 193200ca do not have complex multiplication.Modular form 193200.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.