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Properties

Label 454080.t
Number of curves $6$
Conductor $454080$
CM no
Rank $1$
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 454080.t have rank \(1\).

Complex multiplication

The elliptic curves in class 454080.t do not have complex multiplication.

Modular form 454080.2.a.t

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 454080.t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
454080.t1 454080t4 \([0, -1, 0, -79918081, -274962591839]\) \(215337138023212870452481/234135\) \(61377085440\) \([2]\) \(20971520\) \(2.7368\)  
454080.t2 454080t5 \([0, -1, 0, -12161601, 10420162785]\) \(758850244829023683601/260354661183562275\) \(68250412301303749017600\) \([2]\) \(41943040\) \(3.0834\) \(\Gamma_0(N)\)-optimal*
454080.t3 454080t3 \([0, -1, 0, -5033601, -4225026015]\) \(53804702959424445601/1696325722925625\) \(444681610310615040000\) \([2, 2]\) \(20971520\) \(2.7368\) \(\Gamma_0(N)\)-optimal*
454080.t4 454080t2 \([0, -1, 0, -4994881, -4295039519]\) \(52572582932532371281/54819198225\) \(14370523899494400\) \([2, 2]\) \(10485760\) \(2.3902\) \(\Gamma_0(N)\)-optimal*
454080.t5 454080t1 \([0, -1, 0, -309761, -68124255]\) \(-12539072261612161/414784434735\) \(-108733250859171840\) \([2]\) \(5242880\) \(2.0437\) \(\Gamma_0(N)\)-optimal*
454080.t6 454080t6 \([0, -1, 0, 1474879, -14389970079]\) \(1353482583458377679/342002835319921875\) \(-89653991262105600000000\) \([2]\) \(41943040\) \(3.0834\)  
*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 454080.t1.