Properties

Label 475.a
Number of curves $2$
Conductor $475$
CM no
Rank $1$
Graph

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

Elliptic curves in class 475.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
475.a1 475c2 \([1, -1, 1, -25, 52]\) \(13312053/361\) \(45125\) \([2]\) \(32\) \(-0.32666\)  
475.a2 475c1 \([1, -1, 1, 0, 2]\) \(27/19\) \(-2375\) \([2]\) \(16\) \(-0.67323\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 475.a have rank \(1\).

Complex multiplication

The elliptic curves in class 475.a do not have complex multiplication.

Modular form 475.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} + 4 q^{17} + 3 q^{18} + q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.