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Properties

Label 121680dl
Number of curves $4$
Conductor $121680$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dl1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 121680dl have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(17\) \( 1 - T + 17 T^{2}\) 1.17.ab
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 121680dl do not have complex multiplication.

Modular form 121680.2.a.dl

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 121680dl

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
121680.ca3 121680dl1 \([0, 0, 0, -146523, 138573578]\) \(-24137569/561600\) \(-8094214128343449600\) \([2]\) \(1548288\) \(2.3089\) \(\Gamma_0(N)\)-optimal
121680.ca2 121680dl2 \([0, 0, 0, -5013723, 4301976458]\) \(967068262369/4928040\) \(71026728976213770240\) \([2]\) \(3096576\) \(2.6554\)  
121680.ca4 121680dl3 \([0, 0, 0, 1313637, -3661638838]\) \(17394111071/411937500\) \(-5937162272960256000000\) \([2]\) \(4644864\) \(2.8582\)  
121680.ca1 121680dl4 \([0, 0, 0, -29106363, -57365106838]\) \(189208196468929/10860320250\) \(156527346164324189184000\) \([2]\) \(9289728\) \(3.2048\)