Properties

Label 235200mu
Number of curves $8$
Conductor $235200$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("mu1")
 
E.isogeny_class()
 

Elliptic curves in class 235200mu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.mu7 235200mu1 \([0, -1, 0, 16462367, -19378152863]\) \(1023887723039/928972800\) \(-447662984999731200000000\) \([2]\) \(28311552\) \(3.2255\) \(\Gamma_0(N)\)-optimal
235200.mu6 235200mu2 \([0, -1, 0, -83889633, -173418472863]\) \(135487869158881/51438240000\) \(24787589110824960000000000\) \([2, 2]\) \(56623104\) \(3.5721\)  
235200.mu5 235200mu3 \([0, -1, 0, -591921633, 5419505815137]\) \(47595748626367201/1215506250000\) \(585740676326400000000000000\) \([2, 2]\) \(113246208\) \(3.9187\)  
235200.mu4 235200mu4 \([0, -1, 0, -1181489633, -15626528872863]\) \(378499465220294881/120530818800\) \(58082632912900915200000000\) \([2]\) \(113246208\) \(3.9187\)  
235200.mu2 235200mu5 \([0, -1, 0, -9411921633, 351454565815137]\) \(191342053882402567201/129708022500\) \(62505038393763840000000000\) \([2, 2]\) \(226492416\) \(4.2653\)  
235200.mu8 235200mu6 \([0, -1, 0, 99566367, 17323471735137]\) \(226523624554079/269165039062500\) \(-129708022500000000000000000000\) \([2]\) \(226492416\) \(4.2653\)  
235200.mu1 235200mu7 \([0, -1, 0, -150590721633, 22492949306215137]\) \(783736670177727068275201/360150\) \(173552792985600000000\) \([2]\) \(452984832\) \(4.6118\)  
235200.mu3 235200mu8 \([0, -1, 0, -9353121633, 356062545415137]\) \(-187778242790732059201/4984939585440150\) \(-2402194052249387857305600000000\) \([2]\) \(452984832\) \(4.6118\)  

Rank

sage: E.rank()
 

The elliptic curves in class 235200mu have rank \(0\).

Complex multiplication

The elliptic curves in class 235200mu do not have complex multiplication.

Modular form 235200.2.a.mu

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.