Properties

Label 435344d
Number of curves $2$
Conductor $435344$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 435344d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
435344.d2 435344d1 \([0, 1, 0, 14140, 321244]\) \(253012016/181447\) \(-224207363231488\) \([2]\) \(2211840\) \(1.4420\) \(\Gamma_0(N)\)-optimal*
435344.d1 435344d2 \([0, 1, 0, -63600, 2653444]\) \(5756278756/2705927\) \(13374456624069632\) \([2]\) \(4423680\) \(1.7886\) \(\Gamma_0(N)\)-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 435344d1.

Rank

sage: E.rank()
 

The elliptic curves in class 435344d have rank \(1\).

Complex multiplication

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

Modular form 435344.2.a.d

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - 4q^{5} + q^{7} + q^{9} + 4q^{11} + 8q^{15} - 4q^{17} + O(q^{20})\)  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 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.