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Properties

Label 321398.d
Number of curves $1$
Conductor $321398$
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([1, -1, 0, -1531069681, -23060083371235]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -1531069681, -23060083371235]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -1531069681, -23060083371235]) 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 321398.d1 has rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(7\)\(1 + T\)
\(11\)\(1 + T\)
\(2087\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 3 T + 3 T^{2}\) 1.3.ad
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 - 3 T + 17 T^{2}\) 1.17.ad
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 + 3 T + 23 T^{2}\) 1.23.d
\(29\) \( 1 - 7 T + 29 T^{2}\) 1.29.ah
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 321398.d do not have complex multiplication.

Modular form 321398.2.a.d

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);
 
\(q - q^{2} + 3 q^{3} + q^{4} + 2 q^{5} - 3 q^{6} - q^{7} - q^{8} + 6 q^{9} - 2 q^{10} - q^{11} + 3 q^{12} + 5 q^{13} + q^{14} + 6 q^{15} + q^{16} + 3 q^{17} - 6 q^{18} + 5 q^{19} + O(q^{20})\) 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 321398.d

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
321398.d1 321398d1 \([1, -1, 0, -1531069681, -23060083371235]\) \(-396925117186052884102821926496873/29127157257445988297080832\) \(-29127157257445988297080832\) \([]\) \(356718240\) \(3.9367\) \(\Gamma_0(N)\)-optimal