Properties

Label 65520.b
Number of curves $4$
Conductor $65520$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 65520.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
65520.b1 65520bw4 \([0, 0, 0, -2302803, -1234377198]\) \(16751080718799363/1529437000000\) \(123305609097216000000\) \([2]\) \(1990656\) \(2.5943\)  
65520.b2 65520bw2 \([0, 0, 0, -2249763, -1298832862]\) \(11387025941627437947/10765300\) \(1190556057600\) \([2]\) \(663552\) \(2.0450\)  
65520.b3 65520bw3 \([0, 0, 0, -505683, 116697618]\) \(177381177331203/29679104000\) \(2392775901315072000\) \([2]\) \(995328\) \(2.2477\)  
65520.b4 65520bw1 \([0, 0, 0, -140643, -20284318]\) \(2781982314427707/2703013040\) \(298931618119680\) \([2]\) \(331776\) \(1.6984\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 65520.b have rank \(1\).

Complex multiplication

The elliptic curves in class 65520.b do not have complex multiplication.

Modular form 65520.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} - 6 q^{11} + q^{13} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.