Properties

Label 67760.f
Number of curves $2$
Conductor $67760$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 67760.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67760.f1 67760be2 \([0, -1, 0, -1255638336, 17125985729536]\) \(-249353795628717731809/14000000\) \(-12292195672064000000\) \([]\) \(19160064\) \(3.5775\)  
67760.f2 67760be1 \([0, -1, 0, -15359296, 23949777920]\) \(-456390127585249/17983078400\) \(-15789394177061971558400\) \([]\) \(6386688\) \(3.0282\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 67760.f have rank \(1\).

Complex multiplication

The elliptic curves in class 67760.f do not have complex multiplication.

Modular form 67760.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} - 2 q^{9} - 7 q^{13} + q^{15} - 6 q^{17} - 5 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.