Properties

Label 458640.r
Number of curves $4$
Conductor $458640$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 458640.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.r1 458640r4 \([0, 0, 0, -55343883, -158471897542]\) \(53365044437418169/41984670\) \(14749131929047326720\) \([2]\) \(28311552\) \(2.9847\)  
458640.r2 458640r3 \([0, 0, 0, -8068683, 5334694778]\) \(165369706597369/60703354530\) \(21324968958877196820480\) \([2]\) \(28311552\) \(2.9847\) \(\Gamma_0(N)\)-optimal*
458640.r3 458640r2 \([0, 0, 0, -3482283, -2441087782]\) \(13293525831769/365192100\) \(128291265891713433600\) \([2, 2]\) \(14155776\) \(2.6382\) \(\Gamma_0(N)\)-optimal*
458640.r4 458640r1 \([0, 0, 0, 45717, -124602982]\) \(30080231/19110000\) \(-6713305384181760000\) \([2]\) \(7077888\) \(2.2916\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 458640.r1.

Rank

sage: E.rank()
 

The elliptic curves in class 458640.r have rank \(0\).

Complex multiplication

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

Modular form 458640.2.a.r

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{11} - q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.