Properties

Label 69696eu
Number of curves $2$
Conductor $69696$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("eu1")
 
E.isogeny_class()
 

Elliptic curves in class 69696eu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69696.gu2 69696eu1 \([0, 0, 0, -13068, -106480]\) \(19683/11\) \(137928013774848\) \([2]\) \(245760\) \(1.4034\) \(\Gamma_0(N)\)-optimal
69696.gu1 69696eu2 \([0, 0, 0, -129228, 17782160]\) \(19034163/121\) \(1517208151523328\) \([2]\) \(491520\) \(1.7500\)  

Rank

sage: E.rank()
 

The elliptic curves in class 69696eu have rank \(1\).

Complex multiplication

The elliptic curves in class 69696eu do not have complex multiplication.

Modular form 69696.2.a.eu

sage: E.q_eigenform(10)
 
\(q + 4 q^{5} - 2 q^{7} - 2 q^{13} - 2 q^{17} + 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.