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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 1, 0, -86]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 1, 0, -86]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 1, 0, -86]) 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 1323b have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T + 2 T^{2}\) 1.2.c
\(5\) \( 1 - T + 5 T^{2}\) 1.5.ab
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 - 8 T + 19 T^{2}\) 1.19.ai
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

Each elliptic curve in class 1323b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 1323.2.a.b

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);
 
\(q - 2 q^{4} - 5 q^{13} + 4 q^{16} + 7 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

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

Elliptic curves in class 1323b

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
1323.i3 1323b1 \([0, 0, 1, 0, -86]\) \(0\) \(-3176523\) \([]\) \(126\) \(-0.073509\) \(\Gamma_0(N)\)-optimal \(-3\)
1323.i2 1323b2 \([0, 0, 1, -1470, -21695]\) \(-12288000\) \(-28588707\) \([]\) \(378\) \(0.47580\)   \(-27\)
1323.i4 1323b3 \([0, 0, 1, 0, 2315]\) \(0\) \(-2315685267\) \([]\) \(378\) \(0.47580\)   \(-3\)
1323.i1 1323b4 \([0, 0, 1, -13230, 585758]\) \(-12288000\) \(-20841167403\) \([]\) \(1134\) \(1.0251\)   \(-27\)