Properties

Label 463680ke
Number of curves $4$
Conductor $463680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ke1")
 
E.isogeny_class()
 

Elliptic curves in class 463680ke

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
463680.ke4 463680ke1 \([0, 0, 0, -1963674732, 29202036672944]\) \(4381924769947287308715481/608122186185572352000\) \(116213959551688964730519552000\) \([2]\) \(495452160\) \(4.3036\) \(\Gamma_0(N)\)-optimal*
463680.ke2 463680ke2 \([0, 0, 0, -30287023212, 2028728475306416]\) \(16077778198622525072705635801/388799208512064000000\) \(74300685813099962302464000000\) \([2, 2]\) \(990904320\) \(4.6502\) \(\Gamma_0(N)\)-optimal*
463680.ke1 463680ke3 \([0, 0, 0, -484589558892, 129840204932360624]\) \(65853432878493908038433301506521/38511703125000000\) \(7359701078016000000000000\) \([2]\) \(1981808640\) \(4.9968\) \(\Gamma_0(N)\)-optimal*
463680.ke3 463680ke4 \([0, 0, 0, -29158063212, 2186944090794416]\) \(-14346048055032350809895395801/2509530875136386550792000\) \(-479578818602447875742726356992000\) \([2]\) \(1981808640\) \(4.9968\)  
*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 463680ke1.

Rank

sage: E.rank()
 

The elliptic curves in class 463680ke have rank \(1\).

Complex multiplication

The elliptic curves in class 463680ke do not have complex multiplication.

Modular form 463680.2.a.ke

sage: E.q_eigenform(10)
 
\(q + q^{5} - q^{7} + 4 q^{11} - 6 q^{13} - 2 q^{17} + O(q^{20})\) 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.