Properties

Label 435344.u
Number of curves $2$
Conductor $435344$
CM no
Rank $2$
Graph

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

Elliptic curves in class 435344.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.u1 435344u2 \([0, -1, 0, -2615669, 1630276777]\) \(-1601666731737088/1309814051\) \(-1618488895921474304\) \([]\) \(12192768\) \(2.4230\) \(\Gamma_0(N)\)-optimal*
435344.u2 435344u1 \([0, -1, 0, 34251, 9718201]\) \(3596091392/35177051\) \(-43466984028226304\) \([]\) \(4064256\) \(1.8737\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 435344.u1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344.u have rank \(2\).

Complex multiplication

The elliptic curves in class 435344.u do not have complex multiplication.

Modular form 435344.2.a.u

sage: E.q_eigenform(10)
 
\(q - q^{3} + 3q^{5} + q^{7} - 2q^{9} + 3q^{11} - 3q^{15} - 6q^{17} - 7q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.