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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -6433, 440737]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -6433, 440737]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -6433, 440737]) 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 33600ba have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 - T + 23 T^{2}\) 1.23.ab
\(29\) \( 1 + T + 29 T^{2}\) 1.29.b
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 33600ba do not have complex multiplication.

Modular form 33600.2.a.ba

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);
 
\(q - q^{3} + q^{7} + q^{9} + 4 q^{11} + 6 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

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 33600ba

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
33600.dr5 33600ba1 \([0, -1, 0, -6433, 440737]\) \(-7189057/16128\) \(-66060288000000\) \([2]\) \(98304\) \(1.3404\) \(\Gamma_0(N)\)-optimal
33600.dr4 33600ba2 \([0, -1, 0, -134433, 19000737]\) \(65597103937/63504\) \(260112384000000\) \([2, 2]\) \(196608\) \(1.6870\)  
33600.dr3 33600ba3 \([0, -1, 0, -166433, 9304737]\) \(124475734657/63011844\) \(258096513024000000\) \([2, 2]\) \(393216\) \(2.0335\)  
33600.dr1 33600ba4 \([0, -1, 0, -2150433, 1214488737]\) \(268498407453697/252\) \(1032192000000\) \([2]\) \(393216\) \(2.0335\)  
33600.dr6 33600ba5 \([0, -1, 0, 617567, 71240737]\) \(6359387729183/4218578658\) \(-17279298183168000000\) \([2]\) \(786432\) \(2.3801\)  
33600.dr2 33600ba6 \([0, -1, 0, -1462433, -673687263]\) \(84448510979617/933897762\) \(3825245233152000000\) \([2]\) \(786432\) \(2.3801\)