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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("df1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 87360.df 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 \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 87360.2.a.df

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

Elliptic curves in class 87360.df

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
87360.df1 87360bm4 \([0, -1, 0, -2695425, -113883903]\) \(8261629364934163009/4759323790524030\) \(1247628175743131320320\) \([4]\) \(3932160\) \(2.7375\)  
87360.df2 87360bm2 \([0, -1, 0, -1917825, -1019165823]\) \(2975849362756797409/8263842596100\) \(2166316753512038400\) \([2, 2]\) \(1966080\) \(2.3910\)  
87360.df3 87360bm1 \([0, -1, 0, -1916545, -1020598655]\) \(2969894891179808929/22997520\) \(6028661882880\) \([2]\) \(983040\) \(2.0444\) \(\Gamma_0(N)\)-optimal
87360.df4 87360bm3 \([0, -1, 0, -1160705, -1832766975]\) \(-659704930833045889/5156082432978750\) \(-1351636073310781440000\) \([2]\) \(3932160\) \(2.7375\)