Properties

Label 463680.cb
Number of curves $4$
Conductor $463680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cb1")
 
E.isogeny_class()
 

Elliptic curves in class 463680.cb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.cb1 463680cb4 \([0, 0, 0, -249281868, -1514900635792]\) \(8964546681033941529169/31696875000\) \(6057367142400000000\) \([2]\) \(56623104\) \(3.2463\)  
463680.cb2 463680cb3 \([0, 0, 0, -20771148, -6563498128]\) \(5186062692284555089/2903809817953800\) \(554926697948989410508800\) \([2]\) \(56623104\) \(3.2463\) \(\Gamma_0(N)\)-optimal*
463680.cb3 463680cb2 \([0, 0, 0, -15587148, -23647888528]\) \(2191574502231419089/4115217960000\) \(786430399044648960000\) \([2, 2]\) \(28311552\) \(2.8997\) \(\Gamma_0(N)\)-optimal*
463680.cb4 463680cb1 \([0, 0, 0, -657228, -614007952]\) \(-164287467238609/757170892800\) \(-144697610954656972800\) \([2]\) \(14155776\) \(2.5532\) \(\Gamma_0(N)\)-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 463680.cb1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680.cb have rank \(1\).

Complex multiplication

The elliptic curves in class 463680.cb do not have complex multiplication.

Modular form 463680.2.a.cb

sage: E.q_eigenform(10)
 
\(q - q^{5} - q^{7} + 2 q^{13} + 6 q^{17} + O(q^{20})\) 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 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.