Properties

Label 450840.z
Number of curves $2$
Conductor $450840$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.z1 450840z1 \([0, -1, 0, -81213720, 281308387932]\) \(2396726313900986596/4154072495625\) \(102675672570010250880000\) \([2]\) \(53084160\) \(3.3094\) \(\Gamma_0(N)\)-optimal
450840.z2 450840z2 \([0, -1, 0, -55816400, 460420447500]\) \(-389032340685029858/1627263833203125\) \(-80441739376936797600000000\) \([2]\) \(106168320\) \(3.6559\)  

Rank

sage: E.rank()
 

The elliptic curves in class 450840.z have rank \(1\).

Complex multiplication

The elliptic curves in class 450840.z do not have complex multiplication.

Modular form 450840.2.a.z

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} - 2 q^{11} - q^{13} - q^{15} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.