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Properties

Label 40310.u
Number of curves $2$
Conductor $40310$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 40310.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40310.u1 40310y2 \([1, 1, 1, -2357270, -1711540605]\) \(-1448610774532009781827681/425719930023619660400\) \(-425719930023619660400\) \([]\) \(1536000\) \(2.6729\)  
40310.u2 40310y1 \([1, 1, 1, -19270, 9820995]\) \(-791353021095715681/41277440000000000\) \(-41277440000000000\) \([5]\) \(307200\) \(1.8682\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 40310.u have rank \(1\).

Complex multiplication

The elliptic curves in class 40310.u do not have complex multiplication.

Modular form 40310.2.a.u

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + 3 q^{7} + q^{8} - 2 q^{9} + q^{10} + 2 q^{11} - q^{12} - q^{13} + 3 q^{14} - q^{15} + q^{16} + 3 q^{17} - 2 q^{18} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.