Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, 23, -176]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, 23, -176]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, 23, -176]) 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 285c have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(19\)\(1 + T\)
 
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.ae
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 285c do not have complex multiplication.

Modular form 285.2.a.c

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);
 
\(q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + 4 q^{7} - 3 q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} + 2 q^{13} + 4 q^{14} - q^{15} - q^{16} + 2 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

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 285c

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
285.c4 285c1 \([1, 1, 0, 23, -176]\) \(1256216039/15582375\) \(-15582375\) \([2]\) \(72\) \(0.060992\) \(\Gamma_0(N)\)-optimal
285.c3 285c2 \([1, 1, 0, -382, -2849]\) \(6189976379881/456890625\) \(456890625\) \([2, 2]\) \(144\) \(0.40757\)  
285.c1 285c3 \([1, 1, 0, -6007, -181724]\) \(23977812996389881/146611125\) \(146611125\) \([2]\) \(288\) \(0.75414\)  
285.c2 285c4 \([1, 1, 0, -1237, 13054]\) \(209595169258201/41748046875\) \(41748046875\) \([4]\) \(288\) \(0.75414\)