Properties

Label 13328w
Number of curves $4$
Conductor $13328$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 13328w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13328.e4 13328w1 \([0, 1, 0, -2368, 26676]\) \(3048625/1088\) \(524296650752\) \([2]\) \(13824\) \(0.94971\) \(\Gamma_0(N)\)-optimal
13328.e3 13328w2 \([0, 1, 0, -33728, 2372404]\) \(8805624625/2312\) \(1114130382848\) \([2]\) \(27648\) \(1.2963\)  
13328.e2 13328w3 \([0, 1, 0, -80768, -8860748]\) \(120920208625/19652\) \(9470108254208\) \([2]\) \(41472\) \(1.4990\)  
13328.e1 13328w4 \([0, 1, 0, -88608, -7045004]\) \(159661140625/48275138\) \(23263320926461952\) \([2]\) \(82944\) \(1.8456\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13328w have rank \(2\).

Complex multiplication

The elliptic curves in class 13328w do not have complex multiplication.

Modular form 13328.2.a.w

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.