Properties

Label 405600hd
Number of curves $2$
Conductor $405600$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("hd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 405600hd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.hd2 405600hd1 \([0, 1, 0, -837958, -295246912]\) \(107850176/117\) \(70592081625000000\) \([2]\) \(6021120\) \(2.1486\) \(\Gamma_0(N)\)-optimal*
405600.hd1 405600hd2 \([0, 1, 0, -1049208, -135119412]\) \(26463592/13689\) \(66074188401000000000\) \([2]\) \(12042240\) \(2.4951\) \(\Gamma_0(N)\)-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 405600hd1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600hd have rank \(1\).

Complex multiplication

The elliptic curves in class 405600hd do not have complex multiplication.

Modular form 405600.2.a.hd

sage: E.q_eigenform(10)
 
\(q + q^{3} + 4q^{7} + q^{9} + 2q^{11} - 2q^{19} + O(q^{20})\)  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 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.