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Properties

Label 69366w
Number of curves $2$
Conductor $69366$
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, 0, 430484, 453494672]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, 430484, 453494672]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, 430484, 453494672]) 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 69366w have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(11\)\(1 - T\)
\(1051\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - T + 17 T^{2}\) 1.17.ab
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 7 T + 29 T^{2}\) 1.29.ah
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 69366w do not have complex multiplication.

Modular form 69366.2.a.w

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);
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + q^{14} - q^{15} + q^{16} + 4 q^{17} + q^{18} + 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

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 69366w

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
69366.u2 69366w1 \([1, 0, 0, 430484, 453494672]\) \(8822561460536124355391/93935597925332680704\) \(-93935597925332680704\) \([7]\) \(1893360\) \(2.5135\) \(\Gamma_0(N)\)-optimal
69366.u1 69366w2 \([1, 0, 0, -203025076, -1113860484568]\) \(-925492188434597796818942768449/373958095272087819200664\) \(-373958095272087819200664\) \([]\) \(13253520\) \(3.4864\)