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Properties

Label 451770.t
Number of curves $6$
Conductor $451770$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("t1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 451770.t have rank \(0\).

Complex multiplication

The elliptic curves in class 451770.t do not have complex multiplication.

Modular form 451770.2.a.t

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + 2 q^{13} - q^{15} + q^{16} - 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 451770.t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
451770.t1 451770t6 \([1, 0, 1, -5318594, -4721366074]\) \(6484907238722641/283593750\) \(727623973802343750\) \([2]\) \(13271040\) \(2.5055\)  
451770.t2 451770t3 \([1, 0, 1, -1608604, 785135246]\) \(179415687049201/1443420\) \(3703420813278780\) \([2]\) \(6635520\) \(2.1590\) \(\Gamma_0(N)\)-optimal*
451770.t3 451770t4 \([1, 0, 1, -349124, -65966578]\) \(1834216913521/329422500\) \(845208007968802500\) \([2, 2]\) \(6635520\) \(2.1590\)  
451770.t4 451770t2 \([1, 0, 1, -102704, 11705006]\) \(46694890801/3920400\) \(10058673813843600\) \([2, 2]\) \(3317760\) \(1.8124\) \(\Gamma_0(N)\)-optimal*
451770.t5 451770t1 \([1, 0, 1, 6816, 840622]\) \(13651919/126720\) \(-325128850548480\) \([2]\) \(1658880\) \(1.4658\) \(\Gamma_0(N)\)-optimal*
451770.t6 451770t5 \([1, 0, 1, 677626, -380562778]\) \(13411719834479/32153832150\) \(-82497936297808249350\) \([2]\) \(13271040\) \(2.5055\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 451770.t1.