Properties

Label 53130cr
Number of curves 8
Conductor 53130
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("53130.cs1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 53130cr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53130.cs8 53130cr1 [1, 0, 0, 389490, -79877628] [6] 1327104 \(\Gamma_0(N)\)-optimal
53130.cs7 53130cr2 [1, 0, 0, -2088590, -732603900] [2, 6] 2654208  
53130.cs6 53130cr3 [1, 0, 0, -9421710, -11238222588] [2] 3981312  
53130.cs5 53130cr4 [1, 0, 0, -13968590, 19556060100] [6] 5308416  
53130.cs4 53130cr5 [1, 0, 0, -29857870, -62785836988] [6] 5308416  
53130.cs3 53130cr6 [1, 0, 0, -151146590, -715242391500] [2, 2] 7962624  
53130.cs2 53130cr7 [1, 0, 0, -151545890, -711273429360] [2] 15925248  
53130.cs1 53130cr8 [1, 0, 0, -2418345370, -45774911264488] [2] 15925248  

Rank

sage: E.rank()
 

The elliptic curves in class 53130cr have rank \(1\).

Modular form 53130.2.a.cs

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + 2q^{13} + q^{14} + q^{15} + q^{16} - 6q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.