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Properties

Label 1680t
Number of curves $6$
Conductor $1680$
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 1680t have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1680t do not have complex multiplication.

Modular form 1680.2.a.t

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.

Elliptic curves in class 1680t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1680.q6 1680t1 \([0, 1, 0, 160, 1140]\) \(109902239/188160\) \(-770703360\) \([2]\) \(768\) \(0.39018\) \(\Gamma_0(N)\)-optimal
1680.q5 1680t2 \([0, 1, 0, -1120, 10868]\) \(37966934881/8643600\) \(35404185600\) \([2, 2]\) \(1536\) \(0.73676\)  
1680.q4 1680t3 \([0, 1, 0, -5920, -167692]\) \(5602762882081/345888060\) \(1416757493760\) \([2]\) \(3072\) \(1.0833\)  
1680.q2 1680t4 \([0, 1, 0, -16800, 832500]\) \(128031684631201/9922500\) \(40642560000\) \([2, 4]\) \(3072\) \(1.0833\)  
1680.q1 1680t5 \([0, 1, 0, -268800, 53550900]\) \(524388516989299201/3150\) \(12902400\) \([4]\) \(6144\) \(1.4299\)  
1680.q3 1680t6 \([0, 1, 0, -15680, 949428]\) \(-104094944089921/35880468750\) \(-146966400000000\) \([8]\) \(6144\) \(1.4299\)