Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.k1 450840k2 \([0, -1, 0, -2394410795036, -1425618290334931260]\) \(245689277968779868090419995701456/93342399137270122585475925\) \(576783001549157902219618185606739200\) \([2]\) \(8918138880\) \(5.8281\) \(\Gamma_0(N)\)-optimal*
450840.k2 450840k1 \([0, -1, 0, -127620618411, -29061556541257860]\) \(-595213448747095198927846967296/600281130562949295663181875\) \(-231829235333779159481143252277790000\) \([2]\) \(4459069440\) \(5.4816\) \(\Gamma_0(N)\)-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 450840.k1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 450840.2.a.k

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + 2q^{7} + q^{9} + q^{13} + q^{15} + 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.