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Properties

Label 495495.dt
Number of curves $4$
Conductor $495495$
CM no
Rank $1$
Graph

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 495495.dt have rank \(1\).

Complex multiplication

The elliptic curves in class 495495.dt do not have complex multiplication.

Modular form 495495.2.a.dt

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} + q^{7} - 3 q^{8} + q^{10} - q^{13} + q^{14} - q^{16} + 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 495495.dt

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
495495.dt1 495495dt3 \([1, -1, 0, -18375264, 30322217173]\) \(531301262949272089/4740474375\) \(6122170813177794375\) \([2]\) \(22118400\) \(2.7707\) \(\Gamma_0(N)\)-optimal*
495495.dt2 495495dt2 \([1, -1, 0, -1174509, 451386040]\) \(138742439989609/12224619225\) \(15787704162309104025\) \([2, 2]\) \(11059200\) \(2.4241\) \(\Gamma_0(N)\)-optimal*
495495.dt3 495495dt1 \([1, -1, 0, -254304, -41291717]\) \(1408317602329/242911305\) \(313712169715489545\) \([2]\) \(5529600\) \(2.0775\) \(\Gamma_0(N)\)-optimal*
495495.dt4 495495dt4 \([1, -1, 0, 1302966, 2100888895]\) \(189425802193991/1586486902455\) \(-2048897017758659963895\) \([2]\) \(22118400\) \(2.7707\)  
*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 495495.dt1.