Properties

Label 135240.e
Number of curves $6$
Conductor $135240$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 135240.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
135240.e1 135240bs5 \([0, -1, 0, -69685856, 223928874156]\) \(155324313723954725282/13018359375\) \(3136710578400000000\) \([2]\) \(11010048\) \(2.9916\)  
135240.e2 135240bs4 \([0, -1, 0, -5997616, -5645498564]\) \(198048499826486404/242568272835\) \(29222824684303272960\) \([2]\) \(5505024\) \(2.6450\)  
135240.e3 135240bs3 \([0, -1, 0, -4364936, 3483833340]\) \(76343005935514084/694180580625\) \(83629722757069440000\) \([2, 2]\) \(5505024\) \(2.6450\)  
135240.e4 135240bs6 \([0, -1, 0, -1277936, 8313136140]\) \(-957928673903042/123339801817575\) \(-29718127296585484646400\) \([2]\) \(11010048\) \(2.9916\)  
135240.e5 135240bs2 \([0, -1, 0, -475316, -37050684]\) \(394315384276816/208332909225\) \(6274600559977478400\) \([2, 2]\) \(2752512\) \(2.2985\)  
135240.e6 135240bs1 \([0, -1, 0, 112929, -4579560]\) \(84611246065664/53699121315\) \(-101082366777414960\) \([2]\) \(1376256\) \(1.9519\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 135240.e have rank \(1\).

Complex multiplication

The elliptic curves in class 135240.e do not have complex multiplication.

Modular form 135240.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.