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Properties

Label 68544.t
Number of curves $6$
Conductor $68544$
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 68544.t have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 + T\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 68544.t do not have complex multiplication.

Modular form 68544.2.a.t

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} - q^{7} - 4 q^{11} + 2 q^{13} - q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 68544.t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68544.t1 68544dq6 \([0, 0, 0, -7901915916, -270362650933744]\) \(285531136548675601769470657/17941034271597192\) \(3428585041820215664443392\) \([2]\) \(47185920\) \(4.1682\)  
68544.t2 68544dq4 \([0, 0, 0, -494809356, -4207535177200]\) \(70108386184777836280897/552468975892674624\) \(105578465440762377246081024\) \([2, 2]\) \(23592960\) \(3.8216\)  
68544.t3 68544dq5 \([0, 0, 0, -168539916, -9673331343856]\) \(-2770540998624539614657/209924951154647363208\) \(-40117282902307747339601707008\) \([2]\) \(47185920\) \(4.1682\)  
68544.t4 68544dq2 \([0, 0, 0, -52257036, 36541571600]\) \(82582985847542515777/44772582831427584\) \(8556173822292317630889984\) \([2, 2]\) \(11796480\) \(3.4750\)  
68544.t5 68544dq1 \([0, 0, 0, -40460556, 98935513616]\) \(38331145780597164097/55468445663232\) \(10600185040337928978432\) \([2]\) \(5898240\) \(3.1284\) \(\Gamma_0(N)\)-optimal
68544.t6 68544dq3 \([0, 0, 0, 201551604, 287406031376]\) \(4738217997934888496063/2928751705237796928\) \(-559693166836017780624457728\) \([2]\) \(23592960\) \(3.8216\)