Properties

Label 9408.h
Number of curves $6$
Conductor $9408$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("h1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 9408.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9408.h1 9408q5 \([0, -1, 0, -75329, 7982913]\) \(3065617154/9\) \(138784407552\) \([2]\) \(24576\) \(1.3673\)  
9408.h2 9408q3 \([0, -1, 0, -12609, -540735]\) \(28756228/3\) \(23130734592\) \([2]\) \(12288\) \(1.0208\)  
9408.h3 9408q4 \([0, -1, 0, -4769, 122529]\) \(1556068/81\) \(624529833984\) \([2, 2]\) \(12288\) \(1.0208\)  
9408.h4 9408q2 \([0, -1, 0, -849, -6831]\) \(35152/9\) \(17348050944\) \([2, 2]\) \(6144\) \(0.67418\)  
9408.h5 9408q1 \([0, -1, 0, 131, -755]\) \(2048/3\) \(-361417728\) \([2]\) \(3072\) \(0.32760\) \(\Gamma_0(N)\)-optimal
9408.h6 9408q6 \([0, -1, 0, 3071, 478465]\) \(207646/6561\) \(-101173833105408\) \([2]\) \(24576\) \(1.3673\)  

Rank

sage: E.rank()
 

The elliptic curves in class 9408.h have rank \(2\).

Complex multiplication

The elliptic curves in class 9408.h do not have complex multiplication.

Modular form 9408.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{5} + q^{9} - 4q^{11} - 2q^{13} + 2q^{15} - 2q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.