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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -8, 6]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -8, 6]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -8, 6]) 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 1225h 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
\(3\) \( 1 + 3 T + 3 T^{2}\) 1.3.d
\(11\) \( 1 - T + 11 T^{2}\) 1.11.ab
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + T + 29 T^{2}\) 1.29.b
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 1225.2.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);
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} - 2 q^{9} + q^{12} + 2 q^{13} - q^{16} + 2 q^{17} + 2 q^{18} - 6 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 37 \\ 37 & 1 \end{array}\right)\)

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 1225h

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.b2 1225h1 \([1, 1, 1, -8, 6]\) \(-9317\) \(-6125\) \([]\) \(48\) \(-0.53497\) \(\Gamma_0(N)\)-optimal
1225.b1 1225h2 \([1, 1, 1, -208083, -36621194]\) \(-162677523113838677\) \(-6125\) \([]\) \(1776\) \(1.2705\)