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Properties

Label 458640ii
Number of curves $2$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ii1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640ii have rank \(1\).

Complex multiplication

The elliptic curves in class 458640ii do not have complex multiplication.

Modular form 458640.2.a.ii

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - 4 q^{11} - q^{13} + 6 q^{17} - 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 458640ii

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.ii2 458640ii1 \([0, 0, 0, -3069507, -6471400894]\) \(-26543596087/134784000\) \(-16240828385412513792000\) \([2]\) \(30965760\) \(2.9462\) \(\Gamma_0(N)\)-optimal*
458640.ii1 458640ii2 \([0, 0, 0, -74193987, -245549227966]\) \(374852148636727/760500000\) \(91636618494081024000000\) \([2]\) \(61931520\) \(3.2927\) \(\Gamma_0(N)\)-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 458640ii1.