Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("dt1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 466752.dt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
466752.dt1 466752dt3 \([0, 1, 0, -210894113, 1178543587935]\) \(3957101249824708884951625/772310238681366528\) \(202456495208888147116032\) \([2]\) \(87588864\) \(3.4718\) \(\Gamma_0(N)\)-optimal*
466752.dt2 466752dt4 \([0, 1, 0, -188611873, 1437325066847]\) \(-2830680648734534916567625/1766676274677722124288\) \(-463123585349116788549353472\) \([2]\) \(175177728\) \(3.8184\)  
466752.dt3 466752dt1 \([0, 1, 0, -6420833, -4057229793]\) \(111675519439697265625/37528570137307392\) \(9837889490074308968448\) \([2]\) \(29196288\) \(2.9225\) \(\Gamma_0(N)\)-optimal*
466752.dt4 466752dt2 \([0, 1, 0, 18733727, -28019463649]\) \(2773679829880629422375/2899504554614368272\) \(-760087721964828956295168\) \([2]\) \(58392576\) \(3.2691\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 466752.dt1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 466752.2.a.dt

sage: E.q_eigenform(10)
 
\(q + q^{3} + 4 q^{7} + q^{9} + q^{11} - q^{13} + q^{17} + 2 q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.