Properties

Label 13328p
Number of curves $4$
Conductor $13328$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 13328p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13328.k3 13328p1 \([0, 0, 0, -14651, 671594]\) \(721734273/13328\) \(6422633971712\) \([2]\) \(18432\) \(1.2521\) \(\Gamma_0(N)\)-optimal
13328.k2 13328p2 \([0, 0, 0, -30331, -1018710]\) \(6403769793/2775556\) \(1337513524609024\) \([2, 2]\) \(36864\) \(1.5987\)  
13328.k1 13328p3 \([0, 0, 0, -414491, -102667446]\) \(16342588257633/8185058\) \(3944300087877632\) \([2]\) \(73728\) \(1.9452\)  
13328.k4 13328p4 \([0, 0, 0, 102949, -7549430]\) \(250404380127/196003234\) \(-94452058017243136\) \([2]\) \(73728\) \(1.9452\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13328p have rank \(1\).

Complex multiplication

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

Modular form 13328.2.a.p

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.