Properties

Label 446400.pj
Number of curves $2$
Conductor $446400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("pj1")
 
E.isogeny_class()
 

Elliptic curves in class 446400.pj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446400.pj1 446400pj2 \([0, 0, 0, -539172300, 4495296922000]\) \(5805223604235668521/435937500000000\) \(1301702400000000000000000000\) \([2]\) \(198180864\) \(3.9481\) \(\Gamma_0(N)\)-optimal*
446400.pj2 446400pj1 \([0, 0, 0, 32219700, 311564698000]\) \(1238798620042199/14760960000000\) \(-44075990384640000000000000\) \([2]\) \(99090432\) \(3.6016\) \(\Gamma_0(N)\)-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 446400.pj1.

Rank

sage: E.rank()
 

The elliptic curves in class 446400.pj have rank \(1\).

Complex multiplication

The elliptic curves in class 446400.pj do not have complex multiplication.

Modular form 446400.2.a.pj

sage: E.q_eigenform(10)
 
\(q + 2 q^{7} + 4 q^{11} - 4 q^{13} - 6 q^{17} + O(q^{20})\) 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.