Properties

Label 9408.m
Number of curves $6$
Conductor $9408$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 9408.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9408.m1 9408cd5 \([0, -1, 0, -2458689, -1483076895]\) \(53297461115137/147\) \(4533623980032\) \([2]\) \(98304\) \(2.0869\)  
9408.m2 9408cd3 \([0, -1, 0, -153729, -23115231]\) \(13027640977/21609\) \(666442725064704\) \([2, 2]\) \(49152\) \(1.7403\)  
9408.m3 9408cd4 \([0, -1, 0, -122369, 16417185]\) \(6570725617/45927\) \(1416433663475712\) \([2]\) \(49152\) \(1.7403\)  
9408.m4 9408cd6 \([0, -1, 0, -106689, -37575327]\) \(-4354703137/17294403\) \(-533376327626784768\) \([2]\) \(98304\) \(2.0869\)  
9408.m5 9408cd2 \([0, -1, 0, -12609, -112671]\) \(7189057/3969\) \(122407847460864\) \([2, 2]\) \(24576\) \(1.3937\)  
9408.m6 9408cd1 \([0, -1, 0, 3071, -15455]\) \(103823/63\) \(-1942981705728\) \([2]\) \(12288\) \(1.0472\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 9408.m have rank \(1\).

Complex multiplication

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

Modular form 9408.2.a.m

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.