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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, -1, 0, -25317, 1556716]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -25317, 1556716]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -25317, 1556716]) 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 1575.h have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 1575.h do not have complex multiplication.

Modular form 1575.2.a.h

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);
 
\(q + q^{2} - q^{4} - q^{7} - 3 q^{8} + 6 q^{13} - q^{14} - q^{16} + 2 q^{17} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

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

Elliptic curves in class 1575.h

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
1575.h1 1575f3 \([1, -1, 0, -25317, 1556716]\) \(157551496201/13125\) \(149501953125\) \([2]\) \(3072\) \(1.1881\)  
1575.h2 1575f2 \([1, -1, 0, -1692, 21091]\) \(47045881/11025\) \(125581640625\) \([2, 2]\) \(1536\) \(0.84149\)  
1575.h3 1575f1 \([1, -1, 0, -567, -4784]\) \(1771561/105\) \(1196015625\) \([2]\) \(768\) \(0.49492\) \(\Gamma_0(N)\)-optimal
1575.h4 1575f4 \([1, -1, 0, 3933, 127966]\) \(590589719/972405\) \(-11076300703125\) \([2]\) \(3072\) \(1.1881\)