Properties

Label 466752.bu
Number of curves $4$
Conductor $466752$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bu1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 466752.bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466752.bu1 466752bu3 \([0, -1, 0, -160617801217, 24776466871606177]\) \(1748094148784980747354970849498497/887694600425282263291392\) \(232703813333885193628258664448\) \([2]\) \(1815478272\) \(4.9677\) \(\Gamma_0(N)\)-optimal*
466752.bu2 466752bu4 \([0, -1, 0, -21971478017, -689374449575007]\) \(4474676144192042711273397261697/1806328356954994499451382272\) \(473518140805610078064183154311168\) \([2]\) \(1815478272\) \(4.9677\)  
466752.bu3 466752bu2 \([0, -1, 0, -10092914177, 382734832655265]\) \(433744050935826360922067531137/9612122270219882316693504\) \(2519760180404520830027301912576\) \([2, 2]\) \(907739136\) \(4.6211\) \(\Gamma_0(N)\)-optimal*
466752.bu4 466752bu1 \([0, -1, 0, 57301503, 18331939527585]\) \(79374649975090937760383/553856914190911653543936\) \(-145190266913662344506621558784\) \([2]\) \(453869568\) \(4.2746\) \(\Gamma_0(N)\)-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 466752.bu1.

Rank

sage: E.rank()
 

The elliptic curves in class 466752.bu have rank \(1\).

Complex multiplication

The elliptic curves in class 466752.bu do not have complex multiplication.

Modular form 466752.2.a.bu

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2q^{5} + q^{9} + q^{11} - q^{13} - 2q^{15} - q^{17} + 8q^{19} + O(q^{20})\)  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 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

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

The vertices are labelled with LMFDB labels.