Properties

Label 275.a
Number of curves $4$
Conductor $275$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 275.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
275.a1 275a4 \([1, -1, 1, -1480, 22272]\) \(22930509321/6875\) \(107421875\) \([2]\) \(96\) \(0.51905\)  
275.a2 275a3 \([1, -1, 1, -730, -7228]\) \(2749884201/73205\) \(1143828125\) \([2]\) \(96\) \(0.51905\)  
275.a3 275a2 \([1, -1, 1, -105, 272]\) \(8120601/3025\) \(47265625\) \([2, 2]\) \(48\) \(0.17248\)  
275.a4 275a1 \([1, -1, 1, 20, 22]\) \(59319/55\) \(-859375\) \([4]\) \(24\) \(-0.17409\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 275.a have rank \(1\).

Complex multiplication

The elliptic curves in class 275.a do not have complex multiplication.

Modular form 275.2.a.a

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.