Properties

Label 400710h
Number of curves $4$
Conductor $400710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 400710h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400710.h3 400710h1 \([1, 1, 0, -87008, -9673728]\) \(1548415333009/43192320\) \(2032020746833920\) \([2]\) \(2903040\) \(1.7165\) \(\Gamma_0(N)\)-optimal*
400710.h2 400710h2 \([1, 1, 0, -202528, 21262528]\) \(19528130963089/7116609600\) \(334807168365057600\) \([2, 2]\) \(5806080\) \(2.0631\) \(\Gamma_0(N)\)-optimal*
400710.h1 400710h3 \([1, 1, 0, -2873928, 1873611288]\) \(55799459660732689/15622881480\) \(734992222985183880\) \([2]\) \(11612160\) \(2.4097\) \(\Gamma_0(N)\)-optimal*
400710.h4 400710h4 \([1, 1, 0, 620552, 151144552]\) \(561740261198831/534135885000\) \(-25128893283539685000\) \([2]\) \(11612160\) \(2.4097\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 400710h1.

Rank

sage: E.rank()
 

The elliptic curves in class 400710h have rank \(1\).

Complex multiplication

The elliptic curves in class 400710h do not have complex multiplication.

Modular form 400710.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} + q^{10} - q^{12} - 2 q^{13} - 4 q^{14} + q^{15} + q^{16} + 2 q^{17} - q^{18} + O(q^{20})\) 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.