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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -1537, -23713]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -1537, -23713]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -1537, -23713]) 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 192.d have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 192.d do not have complex multiplication.

Modular form 192.2.a.d

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);
 
\(q + q^{3} + 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} + 2 q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

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 192.d

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
192.d1 192c5 \([0, 1, 0, -1537, -23713]\) \(3065617154/9\) \(1179648\) \([2]\) \(64\) \(0.39437\)  
192.d2 192c4 \([0, 1, 0, -257, 1503]\) \(28756228/3\) \(196608\) \([2]\) \(32\) \(0.047795\)  
192.d3 192c3 \([0, 1, 0, -97, -385]\) \(1556068/81\) \(5308416\) \([2, 2]\) \(32\) \(0.047795\)  
192.d4 192c2 \([0, 1, 0, -17, 15]\) \(35152/9\) \(147456\) \([2, 2]\) \(16\) \(-0.29878\)  
192.d5 192c1 \([0, 1, 0, 3, 3]\) \(2048/3\) \(-3072\) \([2]\) \(8\) \(-0.64535\) \(\Gamma_0(N)\)-optimal
192.d6 192c6 \([0, 1, 0, 63, -1377]\) \(207646/6561\) \(-859963392\) \([4]\) \(64\) \(0.39437\)