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Properties

Label 1680.t
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 1680.t 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 1680.t 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 1680.t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1680.t1 1680j5 \([0, 1, 0, -10480, 409460]\) \(62161150998242/1607445\) \(3292047360\) \([4]\) \(2048\) \(0.93291\)  
1680.t2 1680j3 \([0, 1, 0, -680, 5700]\) \(34008619684/4862025\) \(4978713600\) \([2, 4]\) \(1024\) \(0.58634\)  
1680.t3 1680j2 \([0, 1, 0, -180, -900]\) \(2533446736/275625\) \(70560000\) \([2, 2]\) \(512\) \(0.23977\)  
1680.t4 1680j1 \([0, 1, 0, -175, -952]\) \(37256083456/525\) \(8400\) \([2]\) \(256\) \(-0.10681\) \(\Gamma_0(N)\)-optimal
1680.t5 1680j4 \([0, 1, 0, 240, -4092]\) \(1486779836/8203125\) \(-8400000000\) \([2]\) \(1024\) \(0.58634\)  
1680.t6 1680j6 \([0, 1, 0, 1120, 32340]\) \(75798394558/259416045\) \(-531284060160\) \([4]\) \(2048\) \(0.93291\)