Properties

Label 435344.bd
Number of curves $4$
Conductor $435344$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 435344.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.bd1 435344bd4 \([0, 0, 0, -7546019, -7978568078]\) \(9614292367656708/2093\) \(10344971506688\) \([2]\) \(5505024\) \(2.3217\)  
435344.bd2 435344bd3 \([0, 0, 0, -549419, -80774902]\) \(3710860803108/1577224103\) \(7795670523266382848\) \([2]\) \(5505024\) \(2.3217\) \(\Gamma_0(N)\)-optimal*
435344.bd3 435344bd2 \([0, 0, 0, -471679, -124635810]\) \(9392111857872/4380649\) \(5413006340874496\) \([2, 2]\) \(2752512\) \(1.9751\) \(\Gamma_0(N)\)-optimal*
435344.bd4 435344bd1 \([0, 0, 0, -24674, -2603445]\) \(-21511084032/25465531\) \(-1966676067529264\) \([2]\) \(1376256\) \(1.6285\) \(\Gamma_0(N)\)-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 435344.bd1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344.bd have rank \(1\).

Complex multiplication

The elliptic curves in class 435344.bd do not have complex multiplication.

Modular form 435344.2.a.bd

sage: E.q_eigenform(10)
 
\(q + 2q^{5} - q^{7} - 3q^{9} + 4q^{11} - 2q^{17} + 4q^{19} + O(q^{20})\)  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.