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Properties

Label 87360hk
Number of curves $4$
Conductor $87360$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hk1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 87360hk have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 87360hk do not have complex multiplication.

Modular form 87360.2.a.hk

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} - 4 q^{11} + q^{13} + q^{15} - 2 q^{17} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 87360hk

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
87360.gs3 87360hk1 \([0, 1, 0, -8888969185, -322573631800417]\) \(296304326013275547793071733369/268420373544960000000\) \(70364790402569994240000000\) \([2]\) \(82575360\) \(4.2579\) \(\Gamma_0(N)\)-optimal
87360.gs2 87360hk2 \([0, 1, 0, -8953194465, -317675722032225]\) \(302773487204995438715379645049/8911747415025000000000000\) \(2336161114364313600000000000000\) \([2, 2]\) \(165150720\) \(4.6044\)  
87360.gs4 87360hk3 \([0, 1, 0, 2246805535, -1059409162032225]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-485580822670975979496407040000000\) \([2]\) \(330301440\) \(4.9510\)  
87360.gs1 87360hk4 \([0, 1, 0, -21180798945, 737524970736543]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(373156875000000000000000000000000\) \([2]\) \(330301440\) \(4.9510\)