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Properties

Label 493680.bc
Number of curves $2$
Conductor $493680$
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 493680.bc have rank \(1\).

Complex multiplication

The elliptic curves in class 493680.bc do not have complex multiplication.

Modular form 493680.2.a.bc

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 2 q^{13} + q^{15} - q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 493680.bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
493680.bc1 493680bc1 \([0, -1, 0, -53303081, 149793095256]\) \(590887175978458660864/57171426328125\) \(1620522707156471250000\) \([2]\) \(37632000\) \(3.1057\) \(\Gamma_0(N)\)-optimal
493680.bc2 493680bc2 \([0, -1, 0, -49333676, 173041106460]\) \(-29279123829148431184/11573052978515625\) \(-5248606542764062500000000\) \([2]\) \(75264000\) \(3.4523\)