↑

Properties

Label 404586.be
Number of curves $6$
Conductor $404586$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 404586.be have rank \(0\).

Complex multiplication

The elliptic curves in class 404586.be do not have complex multiplication.

Modular form 404586.2.a.be

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} + 6 q^{11} + q^{14} + q^{16} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 404586.be

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
404586.be1 404586be6 \([1, -1, 0, -14885586702, 699035707564884]\) \(103665426767620308239307625/5961940992\) \(20978542669050150912\) \([2]\) \(358318080\) \(4.0940\) \(\Gamma_0(N)\)-optimal*
404586.be2 404586be5 \([1, -1, 0, -930350862, 10922565624660]\) \(25309080274342544331625/191933498523648\) \(675364800456989110960128\) \([2]\) \(179159040\) \(3.7474\) \(\Gamma_0(N)\)-optimal*
404586.be3 404586be4 \([1, -1, 0, -183942927, 957071907117]\) \(195607431345044517625/752875610010048\) \(2649176355531925547310528\) \([2]\) \(119439360\) \(3.5447\) \(\Gamma_0(N)\)-optimal*
404586.be4 404586be3 \([1, -1, 0, -16997967, -824884371]\) \(154357248921765625/89242711068672\) \(314022232787615242555392\) \([2]\) \(59719680\) \(3.1981\) \(\Gamma_0(N)\)-optimal*
404586.be5 404586be2 \([1, -1, 0, -12077532, -15132932892]\) \(55369510069623625/3916046302812\) \(13779563495806841755932\) \([2]\) \(39813120\) \(2.9954\) \(\Gamma_0(N)\)-optimal*
404586.be6 404586be1 \([1, -1, 0, -11864592, -15726907728]\) \(52492168638015625/293197968\) \(1031688520637877648\) \([2]\) \(19906560\) \(2.6488\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 404586.be1.