Properties

Label 303450gp
Number of curves 8
Conductor 303450
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("303450.gp1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 303450gp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
303450.gp7 303450gp1 [1, 0, 0, -3594588, 2621835792] [2] 10616832 \(\Gamma_0(N)\)-optimal
303450.gp6 303450gp2 [1, 0, 0, -4172588, 1721889792] [2, 2] 21233664  
303450.gp5 303450gp3 [1, 0, 0, -10638963, -10151783583] [2] 31850496  
303450.gp8 303450gp4 [1, 0, 0, 13889912, 12649702292] [2] 42467328  
303450.gp4 303450gp5 [1, 0, 0, -31483088, -66800154708] [2] 42467328  
303450.gp2 303450gp6 [1, 0, 0, -158606963, -768783719583] [2, 2] 63700992  
303450.gp3 303450gp7 [1, 0, 0, -147046963, -885597519583] [2] 127401984  
303450.gp1 303450gp8 [1, 0, 0, -2537654963, -49203821951583] [2] 127401984  

Rank

sage: E.rank()
 

The elliptic curves in class 303450gp have rank \(0\).

Modular form 303450.2.a.gp

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

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.