Properties

Label 464968p
Number of curves $2$
Conductor $464968$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 464968p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
464968.p2 464968p1 \([0, -1, 0, 1513192, -106261540]\) \(7953970437500/4703287687\) \(-226580800339320060928\) \([2]\) \(12579840\) \(2.5952\) \(\Gamma_0(N)\)-optimal*
464968.p1 464968p2 \([0, -1, 0, -6125568, -848749012]\) \(263822189935250/149429406721\) \(14397518001145377179648\) \([2]\) \(25159680\) \(2.9417\) \(\Gamma_0(N)\)-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 464968p1.

Rank

sage: E.rank()
 

The elliptic curves in class 464968p have rank \(1\).

Complex multiplication

The elliptic curves in class 464968p do not have complex multiplication.

Modular form 464968.2.a.p

sage: E.q_eigenform(10)
 
\(q + 2 q^{3} + q^{7} + q^{9} - 6 q^{17} + O(q^{20})\) 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.