Properties

Label 2475.i
Number of curves $4$
Conductor $2475$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 2475.i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2475.i1 2475j3 \([1, -1, 0, -13317, -588034]\) \(22930509321/6875\) \(78310546875\) \([2]\) \(3072\) \(1.0684\)  
2475.i2 2475j4 \([1, -1, 0, -6567, 201716]\) \(2749884201/73205\) \(833850703125\) \([2]\) \(3072\) \(1.0684\)  
2475.i3 2475j2 \([1, -1, 0, -942, -6409]\) \(8120601/3025\) \(34456640625\) \([2, 2]\) \(1536\) \(0.72179\)  
2475.i4 2475j1 \([1, -1, 0, 183, -784]\) \(59319/55\) \(-626484375\) \([2]\) \(768\) \(0.37521\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 2475.i have rank \(1\).

Complex multiplication

The elliptic curves in class 2475.i do not have complex multiplication.

Modular form 2475.2.a.i

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