Properties

Label 12696.r
Number of curves $2$
Conductor $12696$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12696.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.r1 12696r2 \([0, 1, 0, -912, 10080]\) \(3370318/81\) \(2018359296\) \([2]\) \(9216\) \(0.57013\)  
12696.r2 12696r1 \([0, 1, 0, 8, 512]\) \(4/9\) \(-112131072\) \([2]\) \(4608\) \(0.22355\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12696.r have rank \(0\).

Complex multiplication

The elliptic curves in class 12696.r do not have complex multiplication.

Modular form 12696.2.a.r

sage: E.q_eigenform(10)
 
\(q + q^{3} + 2q^{5} - 4q^{7} + q^{9} - 2q^{11} + 6q^{13} + 2q^{15} + 6q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.