Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -1080, 13662]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -1080, 13662]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -1080, 13662]) 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 1728.o have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 7 T + 19 T^{2}\) 1.19.h
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

Each elliptic curve in class 1728.o has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1728.2.a.o

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);
 
\(q + q^{7} - 5 q^{13} - 7 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 27 & 9 & 3 \\ 27 & 1 & 3 & 9 \\ 9 & 3 & 1 & 3 \\ 3 & 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 1728.o

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 CM discriminant
1728.o1 1728v4 \([0, 0, 0, -1080, 13662]\) \(-12288000\) \(-11337408\) \([]\) \(432\) \(0.39872\)   \(-27\)
1728.o2 1728v2 \([0, 0, 0, -120, -506]\) \(-12288000\) \(-15552\) \([]\) \(144\) \(-0.15058\)   \(-27\)
1728.o3 1728v1 \([0, 0, 0, 0, -2]\) \(0\) \(-1728\) \([]\) \(48\) \(-0.69989\) \(\Gamma_0(N)\)-optimal \(-3\)
1728.o4 1728v3 \([0, 0, 0, 0, 54]\) \(0\) \(-1259712\) \([]\) \(144\) \(-0.15058\)   \(-3\)