Properties

Label 422331q
Number of curves $2$
Conductor $422331$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 422331q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422331.q2 422331q1 \([1, 1, 1, 78497, 8729300]\) \(42875/51\) \(-63628046083437927\) \([2]\) \(2875392\) \(1.9106\) \(\Gamma_0(N)\)-optimal*
422331.q1 422331q2 \([1, 1, 1, -459768, 82794564]\) \(8615125/2601\) \(3245030350255334277\) \([2]\) \(5750784\) \(2.2571\) \(\Gamma_0(N)\)-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 422331q1.

Rank

sage: E.rank()
 

The elliptic curves in class 422331q have rank \(1\).

Complex multiplication

The elliptic curves in class 422331q do not have complex multiplication.

Modular form 422331.2.a.q

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} + 2 q^{11} + q^{12} - q^{16} - q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.