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Properties

Label 25230e
Number of curves $4$
Conductor $25230$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -3767697, -616823019]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -3767697, -616823019]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -3767697, -616823019]) 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 25230e have rank \(0\).

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 \(\mathbb{F}_p\)
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 5 T + 17 T^{2}\) 1.17.af
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 - T + 23 T^{2}\) 1.23.ab
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 25230.2.a.e

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{10} - 2 q^{11} - q^{12} + 4 q^{13} + 2 q^{14} - q^{15} + q^{16} + 2 q^{17} - q^{18} + O(q^{20})\) 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 25230e

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.d4 25230e1 \([1, 1, 0, -3767697, -616823019]\) \(9944061759313921/5479747200000\) \(3259481427744451200000\) \([2]\) \(1344000\) \(2.8186\) \(\Gamma_0(N)\)-optimal
25230.d3 25230e2 \([1, 1, 0, -36465777, 84221615349]\) \(9015548596898711041/63863437500000\) \(37987461984225937500000\) \([2]\) \(2688000\) \(3.1651\)  
25230.d2 25230e3 \([1, 1, 0, -1796863797, -29317838120199]\) \(1078651622544688278688321/3692006820\) \(2196091757827049220\) \([2]\) \(6720000\) \(3.6233\)  
25230.d1 25230e4 \([1, 1, 0, -1796889027, -29316973674801]\) \(1078697059648930939019041/63106084995030150\) \(37536971052052102318128150\) \([2]\) \(13440000\) \(3.9699\)