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Properties

Label 156864dd
Number of curves $4$
Conductor $156864$
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, 0, -384033, 91546785]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -384033, 91546785]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -384033, 91546785]) 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 156864dd have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(19\)\(1 - T\)
\(43\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 156864dd do not have complex multiplication.

Modular form 156864.2.a.dd

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);
 
\(q - q^{3} - 4 q^{7} + q^{9} - 2 q^{13} - 6 q^{17} - q^{19} + O(q^{20})\) 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 labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 156864dd

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
156864.q2 156864dd1 \([0, -1, 0, -384033, 91546785]\) \(23894093340015625/55042322688\) \(14429014638723072\) \([2]\) \(1327104\) \(1.9817\) \(\Gamma_0(N)\)-optimal
156864.q3 156864dd2 \([0, -1, 0, -245793, 158206113]\) \(-6264610702863625/37578744274608\) \(-9851042339122839552\) \([2]\) \(2654208\) \(2.3282\)  
156864.q1 156864dd3 \([0, -1, 0, -1752033, -810796959]\) \(2268876641163765625/228097945239552\) \(59794507756877119488\) \([2]\) \(3981312\) \(2.5310\)  
156864.q4 156864dd4 \([0, -1, 0, 2180127, -3935291295]\) \(4371484788393482375/28041364201746432\) \(-7350875377302616670208\) \([2]\) \(7962624\) \(2.8775\)