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Properties

Label 447440bf
Number of curves $4$
Conductor $447440$
CM no
Rank $0$
Graph

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Copy content sage:E = EllipticCurve([0, -1, 0, -15559496, 23191854320]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 447440bf have rank \(0\).

Complex multiplication

The elliptic curves in class 447440bf do not have complex multiplication.

Modular form 447440.2.a.bf

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{3} - q^{5} - q^{7} + q^{9} + 2 q^{13} - 2 q^{15} - q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 447440bf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
447440.bf3 447440bf1 \([0, -1, 0, -15559496, 23191854320]\) \(101706888443185615044169/2155878637210585700\) \(8830478898014559027200\) \([2]\) \(32514048\) \(3.0004\) \(\Gamma_0(N)\)-optimal*
447440.bf4 447440bf2 \([0, -1, 0, 1052184, 70156396016]\) \(31451287726069552151/519132130936329058750\) \(-2126365208315203824640000\) \([2]\) \(65028096\) \(3.3469\)  
447440.bf1 447440bf3 \([0, -1, 0, -1253836536, 17089136410736]\) \(53221428454245552032871153529/531788033000000\) \(2178203783168000000\) \([2]\) \(97542144\) \(3.5497\) \(\Gamma_0(N)\)-optimal*
447440.bf2 447440bf4 \([0, -1, 0, -1253806456, 17089997324400]\) \(-53217598141164952706794620409/5320023923716796875000\) \(-21790817991544000000000000\) \([2]\) \(195084288\) \(3.8962\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 447440bf1.