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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -18449188, 30499006992]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -18449188, 30499006992]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -18449188, 30499006992]) 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 14450w have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 - 3 T + 13 T^{2}\) 1.13.ad
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 - 8 T + 29 T^{2}\) 1.29.ai
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 14450w do not have complex multiplication.

Modular form 14450.2.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);
 
\(q + q^{2} - 2 q^{3} + q^{4} - 2 q^{6} + 2 q^{7} + q^{8} + q^{9} - 6 q^{11} - 2 q^{12} - 2 q^{13} + 2 q^{14} + q^{16} + q^{18} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 14450w

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
14450.v2 14450w1 \([1, 0, 0, -18449188, 30499006992]\) \(1841373668746009/31443200\) \(11858787649700000000\) \([2]\) \(1105920\) \(2.7895\) \(\Gamma_0(N)\)-optimal
14450.v3 14450w2 \([1, 0, 0, -17871188, 32499464992]\) \(-1673672305534489/241375690000\) \(-91034724567150156250000\) \([2]\) \(2211840\) \(3.1361\)  
14450.v1 14450w3 \([1, 0, 0, -30117563, -12516150383]\) \(8010684753304969/4456448000000\) \(1680747204608000000000000\) \([2]\) \(3317760\) \(3.3388\)  
14450.v4 14450w4 \([1, 0, 0, 117850437, -99077430383]\) \(479958568556831351/289000000000000\) \(-108996210015625000000000000\) \([2]\) \(6635520\) \(3.6854\)