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Properties

Label 45675p
Number of curves $2$
Conductor $45675$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 45675p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45675.bg2 45675p1 \([1, -1, 0, -2142, -39609]\) \(-95443993/5887\) \(-67056609375\) \([2]\) \(36864\) \(0.83182\) \(\Gamma_0(N)\)-optimal
45675.bg1 45675p2 \([1, -1, 0, -34767, -2486484]\) \(408023180713/1421\) \(16186078125\) \([2]\) \(73728\) \(1.1784\)  

Rank

sage: E.rank()
 

The elliptic curves in class 45675p have rank \(1\).

Complex multiplication

The elliptic curves in class 45675p do not have complex multiplication.

Modular form 45675.2.a.p

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