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Properties

Label 454597r
Number of curves $2$
Conductor $454597$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 454597r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
454597.r2 454597r1 \([1, -1, 0, -25621349, -49829473776]\) \(43499078731809/82055753\) \(3508800485605708527377\) \([2]\) \(37324800\) \(3.0242\) \(\Gamma_0(N)\)-optimal
454597.r1 454597r2 \([1, -1, 0, -409755814, -3192433531941]\) \(177930109857804849/634933\) \(27150481681974075997\) \([2]\) \(74649600\) \(3.3707\)  

Rank

sage: E.rank()
 

The elliptic curves in class 454597r have rank \(1\).

Complex multiplication

The elliptic curves in class 454597r do not have complex multiplication.

Modular form 454597.2.a.r

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - 4 q^{5} - 2 q^{7} - 3 q^{8} - 3 q^{9} - 4 q^{10} + q^{13} - 2 q^{14} - q^{16} - 3 q^{18} - 8 q^{19} + O(q^{20})\) 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 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.