Properties

Label 225.b
Number of curves $8$
Conductor $225$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 225.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
225.b1 225c7 \([1, -1, 1, -486005, 130530872]\) \(1114544804970241/405\) \(4613203125\) \([2]\) \(768\) \(1.6449\)  
225.b2 225c5 \([1, -1, 1, -30380, 2044622]\) \(272223782641/164025\) \(1868347265625\) \([2, 2]\) \(384\) \(1.2983\)  
225.b3 225c8 \([1, -1, 1, -24755, 2820872]\) \(-147281603041/215233605\) \(-2451645281953125\) \([2]\) \(768\) \(1.6449\)  
225.b4 225c3 \([1, -1, 1, -18005, -925378]\) \(56667352321/15\) \(170859375\) \([2]\) \(192\) \(0.95175\)  
225.b5 225c4 \([1, -1, 1, -2255, 19622]\) \(111284641/50625\) \(576650390625\) \([2, 2]\) \(192\) \(0.95175\)  
225.b6 225c2 \([1, -1, 1, -1130, -14128]\) \(13997521/225\) \(2562890625\) \([2, 2]\) \(96\) \(0.60517\)  
225.b7 225c1 \([1, -1, 1, -5, -628]\) \(-1/15\) \(-170859375\) \([4]\) \(48\) \(0.25860\) \(\Gamma_0(N)\)-optimal
225.b8 225c6 \([1, -1, 1, 7870, 141122]\) \(4733169839/3515625\) \(-40045166015625\) \([2]\) \(384\) \(1.2983\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 225.b do not have complex multiplication.

Modular form 225.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.