Properties

Label 38640.v
Number of curves $4$
Conductor $38640$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("v1")
 
E.isogeny_class()
 

Elliptic curves in class 38640.v

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.v1 38640br4 \([0, -1, 0, -6924496, -7011120704]\) \(8964546681033941529169/31696875000\) \(129830400000000\) \([2]\) \(884736\) \(2.3504\)  
38640.v2 38640br3 \([0, -1, 0, -576976, -30194240]\) \(5186062692284555089/2903809817953800\) \(11894005014338764800\) \([4]\) \(884736\) \(2.3504\)  
38640.v3 38640br2 \([0, -1, 0, -432976, -109336640]\) \(2191574502231419089/4115217960000\) \(16855932764160000\) \([2, 2]\) \(442368\) \(2.0039\)  
38640.v4 38640br1 \([0, -1, 0, -18256, -2836544]\) \(-164287467238609/757170892800\) \(-3101371976908800\) \([2]\) \(221184\) \(1.6573\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 38640.v have rank \(2\).

Complex multiplication

The elliptic curves in class 38640.v do not have complex multiplication.

Modular form 38640.2.a.v

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{7} + q^{9} - 2 q^{13} + q^{15} - 6 q^{17} + O(q^{20})\) 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.