Properties

Label 37440.ef
Number of curves $8$
Conductor $37440$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ef1")
 
E.isogeny_class()
 

Elliptic curves in class 37440.ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
37440.ef1 37440ex8 \([0, 0, 0, -74880012, -249400300016]\) \(242970740812818720001/24375\) \(4658135040000\) \([2]\) \(1572864\) \(2.7784\)  
37440.ef2 37440ex6 \([0, 0, 0, -4680012, -3896860016]\) \(59319456301170001/594140625\) \(113542041600000000\) \([2, 2]\) \(786432\) \(2.4319\)  
37440.ef3 37440ex7 \([0, 0, 0, -4567692, -4092791024]\) \(-55150149867714721/5950927734375\) \(-1137240000000000000000\) \([2]\) \(1572864\) \(2.7784\)  
37440.ef4 37440ex4 \([0, 0, 0, -299532, -57807344]\) \(15551989015681/1445900625\) \(276315912437760000\) \([2, 2]\) \(393216\) \(2.0853\)  
37440.ef5 37440ex2 \([0, 0, 0, -66252, 5551504]\) \(168288035761/27720225\) \(5297417492889600\) \([2, 2]\) \(196608\) \(1.7387\)  
37440.ef6 37440ex1 \([0, 0, 0, -63372, 6140176]\) \(147281603041/5265\) \(1006157168640\) \([2]\) \(98304\) \(1.3921\) \(\Gamma_0(N)\)-optimal
37440.ef7 37440ex3 \([0, 0, 0, 120948, 31235344]\) \(1023887723039/2798036865\) \(-534713171859210240\) \([2]\) \(393216\) \(2.0853\)  
37440.ef8 37440ex5 \([0, 0, 0, 348468, -273720944]\) \(24487529386319/183539412225\) \(-35074927889488281600\) \([2]\) \(786432\) \(2.4319\)  

Rank

sage: E.rank()
 

The elliptic curves in class 37440.ef have rank \(0\).

Complex multiplication

The elliptic curves in class 37440.ef do not have complex multiplication.

Modular form 37440.2.a.ef

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.