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Properties

Label 31680bd
Number of curves $6$
Conductor $31680$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 31680bd have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(11\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 31680bd do not have complex multiplication.

Modular form 31680.2.a.bd

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 31680bd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31680.cx5 31680bd1 \([0, 0, 0, 2868, -229264]\) \(13651919/126720\) \(-24216569118720\) \([2]\) \(49152\) \(1.2494\) \(\Gamma_0(N)\)-optimal
31680.cx4 31680bd2 \([0, 0, 0, -43212, -3196816]\) \(46694890801/3920400\) \(749200107110400\) \([2, 2]\) \(98304\) \(1.5960\)  
31680.cx3 31680bd3 \([0, 0, 0, -146892, 17995376]\) \(1834216913521/329422500\) \(62953620111360000\) \([2, 2]\) \(196608\) \(1.9425\)  
31680.cx2 31680bd4 \([0, 0, 0, -676812, -214312336]\) \(179415687049201/1443420\) \(275841857617920\) \([2]\) \(196608\) \(1.9425\)  
31680.cx6 31680bd5 \([0, 0, 0, 285108, 103876976]\) \(13411719834479/32153832150\) \(-6144693013669478400\) \([2]\) \(393216\) \(2.2891\)  
31680.cx1 31680bd6 \([0, 0, 0, -2237772, 1288414064]\) \(6484907238722641/283593750\) \(54195609600000000\) \([4]\) \(393216\) \(2.2891\)