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Properties

Label 30576.e
Number of curves $1$
Conductor $30576$
CM no
Rank $0$
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, -78779472, -270646784064]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -78779472, -270646784064]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -78779472, -270646784064]) 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 curve 30576.e1 has rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 30576.e do not have complex multiplication.

Modular form 30576.2.a.e

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);
 
\(q - q^{3} - 3 q^{5} + q^{9} - q^{11} + q^{13} + 3 q^{15} - 7 q^{17} + q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny graph

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

Elliptic curves in class 30576.e

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
30576.e1 30576cg1 \([0, -1, 0, -78779472, -270646784064]\) \(-112205650221491190337/745029571313664\) \(-359022526609331224313856\) \([]\) \(5483520\) \(3.3554\) \(\Gamma_0(N)\)-optimal