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Properties

Label 446160be
Number of curves $6$
Conductor $446160$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("be1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 446160be have rank \(1\).

Complex multiplication

The elliptic curves in class 446160be do not have complex multiplication.

Modular form 446160.2.a.be

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + q^{11} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 446160be

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
446160.be5 446160be1 \([0, -1, 0, -7679416, -3567290000]\) \(2533309721804161/1187575234560\) \(23479086409119002787840\) \([2]\) \(30965760\) \(2.9869\) \(\Gamma_0(N)\)-optimal*
446160.be4 446160be2 \([0, -1, 0, -63057336, 190299732336]\) \(1402524686897642881/20523074457600\) \(405753694206418118246400\) \([2, 2]\) \(61931520\) \(3.3334\) \(\Gamma_0(N)\)-optimal*
446160.be2 446160be3 \([0, -1, 0, -1005347256, 12269702674800]\) \(5683972151443376419201/1244117160000\) \(24596954746644234240000\) \([2, 2]\) \(123863040\) \(3.6800\) \(\Gamma_0(N)\)-optimal*
446160.be6 446160be4 \([0, -1, 0, -6814136, 517500172656]\) \(-1769848555063681/5850659851882560\) \(-115671112208406149329059840\) \([2]\) \(123863040\) \(3.6800\)  
446160.be1 446160be5 \([0, -1, 0, -16085555256, 785244972255600]\) \(23281546263261052473907201/1115400\) \(22052138019225600\) \([2]\) \(247726080\) \(4.0266\) \(\Gamma_0(N)\)-optimal*
446160.be3 446160be6 \([0, -1, 0, -1001777976, 12361144772976]\) \(-5623647484692626737921/84122230603125000\) \(-1663147785319379673600000000\) \([2]\) \(247726080\) \(4.0266\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 446160be1.