Properties

Label 12936k
Number of curves $4$
Conductor $12936$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 12936k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12936.u3 12936k1 \([0, 1, 0, -28761644, 59360635872]\) \(87364831012240243408/1760913\) \(53035431305472\) \([2]\) \(552960\) \(2.6169\) \(\Gamma_0(N)\)-optimal
12936.u2 12936k2 \([0, 1, 0, -28762624, 59356387376]\) \(21843440425782779332/3100814593569\) \(373563121785650463744\) \([2, 2]\) \(1105920\) \(2.9635\)  
12936.u1 12936k3 \([0, 1, 0, -31371384, 47944627632]\) \(14171198121996897746/4077720290568771\) \(982506935224576695048192\) \([2]\) \(2211840\) \(3.3101\)  
12936.u4 12936k4 \([0, 1, 0, -26169544, 70496259056]\) \(-8226100326647904626/4152140742401883\) \(-1000438182303414544521216\) \([2]\) \(2211840\) \(3.3101\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12936k have rank \(0\).

Complex multiplication

The elliptic curves in class 12936k do not have complex multiplication.

Modular form 12936.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} + q^{11} + 6 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} + O(q^{20})\) 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.