Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450840.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450840.c1 450840c2 \([0, -1, 0, -456716, 114756516]\) \(1705021456336/68471325\) \(423099220917484800\) \([2]\) \(7077888\) \(2.1485\) \(\Gamma_0(N)\)-optimal*
450840.c2 450840c1 \([0, -1, 0, 12909, 6554916]\) \(615962624/48481875\) \(-18723753648990000\) \([2]\) \(3538944\) \(1.8019\) \(\Gamma_0(N)\)-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.c1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 450840.2.a.c

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.