Properties

Label 53040o
Number of curves $4$
Conductor $53040$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 53040o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53040.bk3 53040o1 \([0, -1, 0, -31335, -2124558]\) \(212670222886967296/616241925\) \(9859870800\) \([2]\) \(98304\) \(1.1477\) \(\Gamma_0(N)\)-optimal
53040.bk2 53040o2 \([0, -1, 0, -31740, -2066400]\) \(13813960087661776/714574355625\) \(182931035040000\) \([2, 2]\) \(196608\) \(1.4943\)  
53040.bk4 53040o3 \([0, -1, 0, 20280, -8225568]\) \(900753985478876/29018422265625\) \(-29714864400000000\) \([4]\) \(393216\) \(1.8409\)  
53040.bk1 53040o4 \([0, -1, 0, -90240, 7808400]\) \(79364416584061444/20404090514925\) \(20893788687283200\) \([4]\) \(393216\) \(1.8409\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53040o have rank \(0\).

Complex multiplication

The elliptic curves in class 53040o do not have complex multiplication.

Modular form 53040.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + 4q^{7} + q^{9} + q^{13} - q^{15} + q^{17} + O(q^{20})\)  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 Cremona 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 Cremona labels.