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Properties

Label 466578.ce
Number of curves $6$
Conductor $466578$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ce1")
 
E.isogeny_class()
 

Elliptic curves in class 466578.ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466578.ce1 466578ce5 \([1, -1, 0, -18461096586, 965455066178280]\) \(54804145548726848737/637608031452\) \(8095367397239575658444728188\) \([2]\) \(830472192\) \(4.5060\) \(\Gamma_0(N)\)-optimal*
466578.ce2 466578ce4 \([1, -1, 0, -4132486206, -102246673939740]\) \(614716917569296417/19093020912\) \(242413852055522757353090928\) \([2]\) \(415236096\) \(4.1594\)  
466578.ce3 466578ce3 \([1, -1, 0, -1183713246, 14262548301252]\) \(14447092394873377/1439452851984\) \(18275961269305379462411722896\) \([2, 2]\) \(415236096\) \(4.1594\) \(\Gamma_0(N)\)-optimal*
466578.ce4 466578ce2 \([1, -1, 0, -269220366, -1454840827308]\) \(169967019783457/26337394944\) \(334391785925818980572027136\) \([2, 2]\) \(207618048\) \(3.8128\) \(\Gamma_0(N)\)-optimal*
466578.ce5 466578ce1 \([1, -1, 0, 29389554, -125608629420]\) \(221115865823/664731648\) \(-8439741417431692697075712\) \([2]\) \(103809024\) \(3.4663\) \(\Gamma_0(N)\)-optimal*
466578.ce6 466578ce6 \([1, -1, 0, 1461784014, 68971960737504]\) \(27207619911317663/177609314617308\) \(-2255010263476052334277230576252\) \([2]\) \(830472192\) \(4.5060\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 466578.ce1.

Rank

sage: E.rank()
 

The elliptic curves in class 466578.ce have rank \(0\).

Complex multiplication

The elliptic curves in class 466578.ce do not have complex multiplication.

Modular form 466578.2.a.ce

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2 q^{5} - q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{13} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.