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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dl1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 87360.dl 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 \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 8 T + 19 T^{2}\) 1.19.i
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 87360.2.a.dl

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{7} + q^{9} + 4 q^{11} - q^{13} - q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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 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.dl

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
87360.dl1 87360fl4 \([0, -1, 0, -10095012065, -390395347739775]\) \(434014578033107719741685694649/103121648659575000\) \(27032721466215628800000\) \([2]\) \(88473600\) \(4.1229\)  
87360.dl2 87360fl2 \([0, -1, 0, -631012065, -6098270939775]\) \(105997782562506306791694649/51649016225625000000\) \(13539479709450240000000000\) \([2, 2]\) \(44236800\) \(3.7763\)  
87360.dl3 87360fl3 \([0, -1, 0, -525880545, -8197053526143]\) \(-61354313914516350666047929/75227254486083984375000\) \(-19720373400000000000000000000\) \([4]\) \(88473600\) \(4.1229\)  
87360.dl4 87360fl1 \([0, -1, 0, -46082785, -60981869183]\) \(41285728533151645510969/17760741842188800000\) \(4655871909478740787200000\) \([2]\) \(22118400\) \(3.4297\) \(\Gamma_0(N)\)-optimal