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Properties

Label 1225.i
Number of curves $2$
Conductor $1225$
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, 1, -181708, -29902307]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 1, -181708, -29902307]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 1, -181708, -29902307]) 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 1225.i have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - 2 T + 2 T^{2}\) 1.2.ac
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 + T + 13 T^{2}\) 1.13.b
\(17\) \( 1 + 7 T + 17 T^{2}\) 1.17.h
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1225.i do not have complex multiplication.

Modular form 1225.2.a.i

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);
 
\(q + 2 q^{2} - q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{9} - 3 q^{11} - 2 q^{12} - q^{13} - 4 q^{16} - 7 q^{17} - 4 q^{18} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

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 1225.i

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
1225.i1 1225i2 \([0, -1, 1, -181708, -29902307]\) \(-2887553024/16807\) \(-3861966294921875\) \([]\) \(9600\) \(1.8319\)  
1225.i2 1225i1 \([0, -1, 1, 2042, 48943]\) \(4096/7\) \(-1608482421875\) \([]\) \(1920\) \(1.0271\) \(\Gamma_0(N)\)-optimal