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Properties

Label 458640.bc
Number of curves $4$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640.bc have rank \(1\).

Complex multiplication

The elliptic curves in class 458640.bc do not have complex multiplication.

Modular form 458640.2.a.bc

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{11} + q^{13} - 2 q^{17} + 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 458640.bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.bc1 458640bc3 \([0, 0, 0, -2335183083723, 853785517389683578]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(500065901456748046875000000000000000000\) \([2]\) \(15854469120\) \(6.1267\) \(\Gamma_0(N)\)-optimal*
458640.bc2 458640bc2 \([0, 0, 0, -987089689803, -367745902903640198]\) \(302773487204995438715379645049/8911747415025000000000000\) \(3130679325157258723430400000000000000\) \([2, 2]\) \(7927234560\) \(5.7801\) \(\Gamma_0(N)\)-optimal*
458640.bc3 458640bc1 \([0, 0, 0, -980008852683, -373415870481973382]\) \(296304326013275547793071733369/268420373544960000000\) \(94295548871975888077455360000000\) \([2]\) \(3963617280\) \(5.4336\) \(\Gamma_0(N)\)-optimal*
458640.bc4 458640bc4 \([0, 0, 0, 247710310197, -1226399398183640198]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-650724743632476078837224457154560000000\) \([2]\) \(15854469120\) \(6.1267\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 458640.bc1.