Properties

Label 7056.k
Number of curves $6$
Conductor $7056$
CM no
Rank $0$
Graph

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

Elliptic curves in class 7056.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7056.k1 7056by3 \([0, 0, 0, -9483411, -11240741806]\) \(268498407453697/252\) \(88527103967232\) \([2]\) \(147456\) \(2.4045\)  
7056.k2 7056by5 \([0, 0, 0, -6449331, 6243532274]\) \(84448510979617/933897762\) \(328076445521187643392\) \([2]\) \(294912\) \(2.7511\)  
7056.k3 7056by4 \([0, 0, 0, -733971, -85657390]\) \(124475734657/63011844\) \(22135936765694459904\) \([2, 2]\) \(147456\) \(2.4045\)  
7056.k4 7056by2 \([0, 0, 0, -592851, -175550830]\) \(65597103937/63504\) \(22308830199742464\) \([2, 2]\) \(73728\) \(2.0579\)  
7056.k5 7056by1 \([0, 0, 0, -28371, -4061806]\) \(-7189057/16128\) \(-5665734653902848\) \([2]\) \(36864\) \(1.7114\) \(\Gamma_0(N)\)-optimal
7056.k6 7056by6 \([0, 0, 0, 2723469, -661666894]\) \(6359387729183/4218578658\) \(-1481978378772666851328\) \([2]\) \(294912\) \(2.7511\)  

Rank

sage: E.rank()
 

The elliptic curves in class 7056.k have rank \(0\).

Complex multiplication

The elliptic curves in class 7056.k do not have complex multiplication.

Modular form 7056.2.a.k

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

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.