Properties

Label 394944du
Number of curves $4$
Conductor $394944$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("du1")
 
E.isogeny_class()
 

Elliptic curves in class 394944du

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
394944.du4 394944du1 \([0, -1, 0, -51625537, 107426494177]\) \(32765849647039657/8229948198912\) \(3822021577595352942379008\) \([2]\) \(61931520\) \(3.4269\) \(\Gamma_0(N)\)-optimal
394944.du2 394944du2 \([0, -1, 0, -767790657, 8188204009185]\) \(107784459654566688937/10704361149504\) \(4971149064241533614555136\) \([2, 2]\) \(123863040\) \(3.7734\)  
394944.du1 394944du3 \([0, -1, 0, -12284357697, 524059004685537]\) \(441453577446719855661097/4354701912\) \(2022341364658898731008\) \([2]\) \(247726080\) \(4.1200\)  
394944.du3 394944du4 \([0, -1, 0, -709865537, 9475705651425]\) \(-85183593440646799657/34223681512621656\) \(-15893617559255523924381794304\) \([2]\) \(247726080\) \(4.1200\)  

Rank

sage: E.rank()
 

The elliptic curves in class 394944du have rank \(0\).

Complex multiplication

The elliptic curves in class 394944du do not have complex multiplication.

Modular form 394944.2.a.du

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

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.