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Properties

Label 13475i
Number of curves $2$
Conductor $13475$
CM no
Rank $1$
Graph

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

Elliptic curves in class 13475i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13475.g2 13475i1 \([1, 1, 1, 4262, -404594]\) \(4657463/41503\) \(-76293538234375\) \([2]\) \(36864\) \(1.3451\) \(\Gamma_0(N)\)-optimal
13475.g1 13475i2 \([1, 1, 1, -63113, -5659844]\) \(15124197817/1294139\) \(2378971237671875\) \([2]\) \(73728\) \(1.6917\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13475i have rank \(1\).

Complex multiplication

The elliptic curves in class 13475i do not have complex multiplication.

Modular form 13475.2.a.i

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} - q^{4} - 2 q^{6} + 3 q^{8} + q^{9} + q^{11} - 2 q^{12} + 4 q^{13} - q^{16} + 4 q^{17} - q^{18} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.