Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -567, 837216]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -567, 837216]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -567, 837216]) 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 curves in class 27225bi have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 27225bi do not have complex multiplication.

Modular form 27225.2.a.bi

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);
 
\(q + q^{2} - q^{4} - 3 q^{8} - 2 q^{13} - q^{16} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labeled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 27225bi

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
27225.bp7 27225bi1 \([1, -1, 0, -567, 837216]\) \(-1/15\) \(-302687805234375\) \([2]\) \(61440\) \(1.4575\) \(\Gamma_0(N)\)-optimal
27225.bp6 27225bi2 \([1, -1, 0, -136692, 19214091]\) \(13997521/225\) \(4540317078515625\) \([2, 2]\) \(122880\) \(1.8041\)  
27225.bp5 27225bi3 \([1, -1, 0, -272817, -25298784]\) \(111284641/50625\) \(1021571342666015625\) \([2, 2]\) \(245760\) \(2.1507\)  
27225.bp4 27225bi4 \([1, -1, 0, -2178567, 1238213466]\) \(56667352321/15\) \(302687805234375\) \([2]\) \(245760\) \(2.1507\)  
27225.bp8 27225bi5 \([1, -1, 0, 952308, -190690659]\) \(4733169839/3515625\) \(-70942454351806640625\) \([2]\) \(491520\) \(2.4973\)  
27225.bp2 27225bi6 \([1, -1, 0, -3675942, -2710364409]\) \(272223782641/164025\) \(3309891150237890625\) \([2, 2]\) \(491520\) \(2.4973\)  
27225.bp3 27225bi7 \([1, -1, 0, -2995317, -3745595034]\) \(-147281603041/215233605\) \(-4343239167342160078125\) \([2]\) \(983040\) \(2.8438\)  
27225.bp1 27225bi8 \([1, -1, 0, -58806567, -173560171284]\) \(1114544804970241/405\) \(8172570741328125\) \([2]\) \(983040\) \(2.8438\)