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Properties

Label 485520.ca
Number of curves $6$
Conductor $485520$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ca1")
 
E.isogeny_class()
 

Elliptic curves in class 485520.ca

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485520.ca1 485520ca5 \([0, -1, 0, -7051696, 7208156320]\) \(784478485879202/221484375\) \(10948801298400000000\) \([2]\) \(18874368\) \(2.6344\) \(\Gamma_0(N)\)-optimal*
485520.ca2 485520ca3 \([0, -1, 0, -497176, 82082176]\) \(549871953124/200930625\) \(4966376268954240000\) \([2, 2]\) \(9437184\) \(2.2878\) \(\Gamma_0(N)\)-optimal*
485520.ca3 485520ca2 \([0, -1, 0, -213956, -37096800]\) \(175293437776/4862025\) \(30043510762809600\) \([2, 2]\) \(4718592\) \(1.9412\) \(\Gamma_0(N)\)-optimal*
485520.ca4 485520ca1 \([0, -1, 0, -212511, -37636074]\) \(2748251600896/2205\) \(851573434320\) \([2]\) \(2359296\) \(1.5946\) \(\Gamma_0(N)\)-optimal*
485520.ca5 485520ca4 \([0, -1, 0, 46144, -121785360]\) \(439608956/259416045\) \(-6411952830356075520\) \([2]\) \(9437184\) \(2.2878\)  
485520.ca6 485520ca6 \([0, -1, 0, 1525824, 578930976]\) \(7947184069438/7533176175\) \(-372393082292671641600\) \([2]\) \(18874368\) \(2.6344\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 485520.ca1.

Rank

sage: E.rank()
 

The elliptic curves in class 485520.ca have rank \(0\).

Complex multiplication

The elliptic curves in class 485520.ca do not have complex multiplication.

Modular form 485520.2.a.ca

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.