Properties

Label 193200ca
Number of curves $4$
Conductor $193200$
CM no
Rank $0$
Graph

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Show commands: 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)
 
\(q + q^{3} - q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.