Properties

Label 88445p
Number of curves $4$
Conductor $88445$
CM no
Rank $1$
Graph

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

Elliptic curves in class 88445p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
88445.k3 88445p1 \([1, -1, 1, -250963, -48239838]\) \(315821241/665\) \(3680709067756385\) \([2]\) \(552960\) \(1.8719\) \(\Gamma_0(N)\)-optimal
88445.k2 88445p2 \([1, -1, 1, -339408, -11163694]\) \(781229961/442225\) \(2447671530057996025\) \([2, 2]\) \(1105920\) \(2.2184\)  
88445.k4 88445p3 \([1, -1, 1, 1341047, -89808988]\) \(48188806119/28511875\) \(-157810401280055006875\) \([2]\) \(2211840\) \(2.5650\)  
88445.k1 88445p4 \([1, -1, 1, -3434983, 2439293476]\) \(809818183161/4561235\) \(25245983495741044715\) \([2]\) \(2211840\) \(2.5650\)  

Rank

sage: E.rank()
 

The elliptic curves in class 88445p have rank \(1\).

Complex multiplication

The elliptic curves in class 88445p do not have complex multiplication.

Modular form 88445.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - q^{5} + 3 q^{8} - 3 q^{9} + q^{10} - 4 q^{11} - 2 q^{13} - q^{16} - 2 q^{17} + 3 q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.