Properties

Label 137904cv
Number of curves $4$
Conductor $137904$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("cv1")
 
E.isogeny_class()
 

Elliptic curves in class 137904cv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
137904.f3 137904cv1 \([0, -1, 0, -8844, -316512]\) \(61918288/153\) \(189056454912\) \([2]\) \(245760\) \(1.0416\) \(\Gamma_0(N)\)-optimal
137904.f2 137904cv2 \([0, -1, 0, -12224, -48816]\) \(40873252/23409\) \(115702550406144\) \([2, 2]\) \(491520\) \(1.3882\)  
137904.f1 137904cv3 \([0, -1, 0, -127144, 17419024]\) \(22994537186/111537\) \(1102577245046784\) \([2]\) \(983040\) \(1.7348\)  
137904.f4 137904cv4 \([0, -1, 0, 48616, -438192]\) \(1285471294/751689\) \(-7430674903861248\) \([2]\) \(983040\) \(1.7348\)  

Rank

sage: E.rank()
 

The elliptic curves in class 137904cv have rank \(2\).

Complex multiplication

The elliptic curves in class 137904cv do not have complex multiplication.

Modular form 137904.2.a.cv

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + 4 q^{11} + 2 q^{15} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.