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Properties

Label 81840bz
Number of curves $4$
Conductor $81840$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -7786800, -8359184448]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -7786800, -8359184448]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -7786800, -8359184448]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 81840bz have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(11\)\(1 + T\)
\(31\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 81840bz do not have complex multiplication.

Modular form 81840.2.a.bz

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{3} + q^{5} + 2 q^{7} + q^{9} - q^{11} - 6 q^{13} - q^{15} + 8 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 81840bz

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
81840.bl3 81840bz1 \([0, -1, 0, -7786800, -8359184448]\) \(12747965531857798561201/2986780262400000\) \(12233851954790400000\) \([2]\) \(4224000\) \(2.6530\) \(\Gamma_0(N)\)-optimal
81840.bl4 81840bz2 \([0, -1, 0, -6885680, -10368321600]\) \(-8814635019030000319921/6242069790000000000\) \(-25567517859840000000000\) \([2]\) \(8448000\) \(2.9996\)  
81840.bl1 81840bz3 \([0, -1, 0, -139710000, 634323936192]\) \(73628549562506871957390001/178215946908754500240\) \(729972518538258432983040\) \([2]\) \(21120000\) \(3.4577\)  
81840.bl2 81840bz4 \([0, -1, 0, -88173680, 1108169476800]\) \(-18508902577171306222471921/118801759721890483665900\) \(-486612007820863421095526400\) \([2]\) \(42240000\) \(3.8043\)