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Properties

Label 25230.h
Number of curves $4$
Conductor $25230$
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, -168479, 26272946]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -168479, 26272946]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -168479, 26272946]) 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 25230.h have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(29\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 25230.h do not have complex multiplication.

Modular form 25230.2.a.h

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);
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 2 q^{7} - q^{8} + q^{9} + q^{10} + q^{12} + 6 q^{13} + 2 q^{14} - q^{15} + q^{16} + 2 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 25230.h

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
25230.h1 25230i3 \([1, 0, 1, -168479, 26272946]\) \(21685195471991381/309586821120\) \(7550512980295680\) \([2]\) \(224000\) \(1.8511\)  
25230.h2 25230i4 \([1, 0, 1, -19999, 70995122]\) \(-36267977929301/89261680665600\) \(-2177003129753318400\) \([2]\) \(448000\) \(2.1977\)  
25230.h3 25230i1 \([1, 0, 1, -16954, -851044]\) \(22095784790981/450000\) \(10975050000\) \([2]\) \(44800\) \(1.0464\) \(\Gamma_0(N)\)-optimal
25230.h4 25230i2 \([1, 0, 1, -16374, -911828]\) \(-19904714311301/3164062500\) \(-77168320312500\) \([2]\) \(89600\) \(1.3929\)