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Properties

Label 366912bh
Number of curves $3$
Conductor $366912$
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, 0, 0, 380436, -858833584]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, 380436, -858833584]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, 380436, -858833584]) 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 366912bh have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 3 T + 5 T^{2}\) 1.5.d
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - 7 T + 19 T^{2}\) 1.19.ah
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 + 9 T + 29 T^{2}\) 1.29.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 366912bh do not have complex multiplication.

Modular form 366912.2.a.bh

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);
 
\(q - 3 q^{5} - 3 q^{11} + q^{13} - 3 q^{17} + 7 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

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 366912bh

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
366912.bh3 366912bh1 \([0, 0, 0, 380436, -858833584]\) \(270840023/14329224\) \(-322165003890223742976\) \([]\) \(15925248\) \(2.6148\) \(\Gamma_0(N)\)-optimal
366912.bh2 366912bh2 \([0, 0, 0, -3429804, 23412395216]\) \(-198461344537/10417365504\) \(-234214399755495665565696\) \([]\) \(47775744\) \(3.1641\)  
366912.bh1 366912bh3 \([0, 0, 0, -735419244, 7676400516176]\) \(-1956469094246217097/36641439744\) \(-823812202089182140563456\) \([]\) \(143327232\) \(3.7134\)