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Properties

Label 444360.u
Number of curves $4$
Conductor $444360$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("u1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 444360.u have rank \(1\).

Complex multiplication

The elliptic curves in class 444360.u do not have complex multiplication.

Modular form 444360.2.a.u

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - q^{7} + q^{9} - 2 q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 444360.u

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
444360.u1 444360u3 \([0, -1, 0, -1976520, 1070203932]\) \(5633270409316/14175\) \(2148770536012800\) \([2]\) \(6307840\) \(2.1802\) \(\Gamma_0(N)\)-optimal*
444360.u2 444360u4 \([0, -1, 0, -347200, -57522500]\) \(30534944836/8203125\) \(1243501467600000000\) \([2]\) \(6307840\) \(2.1802\)  
444360.u3 444360u2 \([0, -1, 0, -125020, 16330132]\) \(5702413264/275625\) \(10445412327840000\) \([2, 2]\) \(3153920\) \(1.8336\) \(\Gamma_0(N)\)-optimal*
444360.u4 444360u1 \([0, -1, 0, 4585, 984900]\) \(4499456/180075\) \(-426521003386800\) \([2]\) \(1576960\) \(1.4871\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 444360.u1.