Properties

Label 13680be
Number of curves $4$
Conductor $13680$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("be1")
 
E.isogeny_class()
 

Elliptic curves in class 13680be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13680.y4 13680be1 \([0, 0, 0, 1344237, -1275542062]\) \(89962967236397039/287450726400000\) \(-858323269818777600000\) \([2]\) \(460800\) \(2.7000\) \(\Gamma_0(N)\)-optimal
13680.y3 13680be2 \([0, 0, 0, -12664083, -14944860718]\) \(75224183150104868881/11219310000000000\) \(33500680151040000000000\) \([2]\) \(921600\) \(3.0466\)  
13680.y2 13680be3 \([0, 0, 0, -475410963, -3989808404782]\) \(-3979640234041473454886161/1471455901872240\) \(-4393743779696078684160\) \([2]\) \(2304000\) \(3.5047\)  
13680.y1 13680be4 \([0, 0, 0, -7606576083, -255347690321518]\) \(16300610738133468173382620881/2228489100\) \(6654232796774400\) \([2]\) \(4608000\) \(3.8513\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13680be have rank \(0\).

Complex multiplication

The elliptic curves in class 13680be do not have complex multiplication.

Modular form 13680.2.a.be

sage: E.q_eigenform(10)
 
\(q - q^{5} + 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.