Properties

Label 225c
Number of curves 8
Conductor 225
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("225.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 225c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
225.b7 225c1 [1, -1, 1, -5, -628] [4] 48 \(\Gamma_0(N)\)-optimal
225.b6 225c2 [1, -1, 1, -1130, -14128] [2, 2] 96  
225.b4 225c3 [1, -1, 1, -18005, -925378] [2] 192  
225.b5 225c4 [1, -1, 1, -2255, 19622] [2, 2] 192  
225.b2 225c5 [1, -1, 1, -30380, 2044622] [2, 2] 384  
225.b8 225c6 [1, -1, 1, 7870, 141122] [2] 384  
225.b1 225c7 [1, -1, 1, -486005, 130530872] [2] 768  
225.b3 225c8 [1, -1, 1, -24755, 2820872] [2] 768  

Rank

sage: E.rank()
 

The elliptic curves in class 225c have rank \(0\).

Modular form 225.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{4} + 3q^{8} + 4q^{11} + 2q^{13} - q^{16} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.