Properties

Label 471510s
Number of curves $2$
Conductor $471510$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 471510s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
471510.s2 471510s1 \([1, -1, 0, -2649360, 1730540800]\) \(-15780576012359283/787251200000\) \(-102597601790361600000\) \([2]\) \(16588800\) \(2.6009\) \(\Gamma_0(N)\)-optimal*
471510.s1 471510s2 \([1, -1, 0, -42884880, 108105208576]\) \(66928707375050045043/155000000000\) \(20200195665000000000\) \([2]\) \(33177600\) \(2.9474\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 471510s1.

Rank

sage: E.rank()
 

The elliptic curves in class 471510s have rank \(0\).

Complex multiplication

The elliptic curves in class 471510s do not have complex multiplication.

Modular form 471510.2.a.s

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} + 2 q^{11} + q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.