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Properties

Label 112530bf
Number of curves $4$
Conductor $112530$
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([1, 0, 1, -58887678, -173903801744]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -58887678, -173903801744]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -58887678, -173903801744]) 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 112530bf have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 112530bf do not have complex multiplication.

Modular form 112530.2.a.bf

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);
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 6 q^{13} - 2 q^{14} + q^{15} + q^{16} - 8 q^{17} - q^{18} + O(q^{20})\) 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 112530bf

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
112530.bg3 112530bf1 \([1, 0, 1, -58887678, -173903801744]\) \(12747965531857798561201/2986780262400000\) \(5291263428437606400000\) \([2]\) \(21120000\) \(3.1588\) \(\Gamma_0(N)\)-optimal
112530.bg4 112530bf2 \([1, 0, 1, -52072958, -215680761232]\) \(-8814635019030000319921/6242069790000000000\) \(-11058207399242190000000000\) \([2]\) \(42240000\) \(3.5054\)  
112530.bg1 112530bf3 \([1, 0, 1, -1056556878, 13190899053616]\) \(73628549562506871957390001/178215946908754500240\) \(315720421121620031199674640\) \([2]\) \(105600000\) \(3.9635\)  
112530.bg2 112530bf4 \([1, 0, 1, -666813458, 23045795274368]\) \(-18508902577171306222471921/118801759721890483665900\) \(-210464564254672027133645469900\) \([2]\) \(211200000\) \(4.3101\)