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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -27744, -1781010]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -27744, -1781010]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -27744, -1781010]) 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 102.c have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 102.c do not have complex multiplication.

Modular form 102.2.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);
 
\(q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + q^{8} + q^{9} - 2 q^{10} - 4 q^{11} + q^{12} - 2 q^{13} - 2 q^{15} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

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

Elliptic curves in class 102.c

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
102.c1 102b5 \([1, 0, 0, -27744, -1781010]\) \(2361739090258884097/5202\) \(5202\) \([2]\) \(128\) \(0.84708\)  
102.c2 102b3 \([1, 0, 0, -1734, -27936]\) \(576615941610337/27060804\) \(27060804\) \([2, 2]\) \(64\) \(0.50050\)  
102.c3 102b6 \([1, 0, 0, -1644, -30942]\) \(-491411892194497/125563633938\) \(-125563633938\) \([2]\) \(128\) \(0.84708\)  
102.c4 102b2 \([1, 0, 0, -114, -396]\) \(163936758817/30338064\) \(30338064\) \([2, 4]\) \(32\) \(0.15393\)  
102.c5 102b1 \([1, 0, 0, -34, 68]\) \(4354703137/352512\) \(352512\) \([8]\) \(16\) \(-0.19264\) \(\Gamma_0(N)\)-optimal
102.c6 102b4 \([1, 0, 0, 226, -2232]\) \(1276229915423/2927177028\) \(-2927177028\) \([4]\) \(64\) \(0.50050\)