Properties

Label 12696s
Number of curves $2$
Conductor $12696$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12696s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.p2 12696s1 \([0, 1, 0, -7444, 245600]\) \(-14647977776/59049\) \(-183922990848\) \([2]\) \(23040\) \(1.0181\) \(\Gamma_0(N)\)-optimal
12696.p1 12696s2 \([0, 1, 0, -119224, 15805376]\) \(15043017316604/243\) \(3027538944\) \([2]\) \(46080\) \(1.3646\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12696s have rank \(2\).

Complex multiplication

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

Modular form 12696.2.a.s

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} - 2q^{7} + q^{9} - 4q^{11} - 6q^{13} - 2q^{15} - 6q^{17} - 6q^{19} + O(q^{20})\)  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 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.