Properties

Label 323400.cm
Number of curves $4$
Conductor $323400$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cm1")
 
E.isogeny_class()
 

Elliptic curves in class 323400.cm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
323400.cm1 323400cm4 \([0, -1, 0, -784284608, 5994647023212]\) \(14171198121996897746/4077720290568771\) \(15351670862884010860128000000\) \([2]\) \(283115520\) \(4.1148\)  
323400.cm2 323400cm2 \([0, -1, 0, -719065608, 7420986553212]\) \(21843440425782779332/3100814593569\) \(5836923777900788496000000\) \([2, 2]\) \(141557760\) \(3.7682\)  
323400.cm3 323400cm1 \([0, -1, 0, -719041108, 7421517566212]\) \(87364831012240243408/1760913\) \(828678614148000000\) \([2]\) \(70778880\) \(3.4217\) \(\Gamma_0(N)\)-optimal
323400.cm4 323400cm3 \([0, -1, 0, -654238608, 8813340859212]\) \(-8226100326647904626/4152140742401883\) \(-15631846598490852258144000000\) \([2]\) \(283115520\) \(4.1148\)  

Rank

sage: E.rank()
 

The elliptic curves in class 323400.cm have rank \(0\).

Complex multiplication

The elliptic curves in class 323400.cm do not have complex multiplication.

Modular form 323400.2.a.cm

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + q^{11} - 6 q^{13} - 2 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.