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Properties

Label 176400.ch
Number of curves $8$
Conductor $176400$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -381027675, 2862746514250]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -381027675, 2862746514250]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -381027675, 2862746514250]) 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 176400.ch have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 176400.ch do not have complex multiplication.

Modular form 176400.2.a.ch

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);
 
\(q - 4 q^{11} - 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

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 176400.ch

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
176400.ch1 176400dv7 \([0, 0, 0, -381027675, 2862746514250]\) \(1114544804970241/405\) \(2223057856320000000\) \([2]\) \(18874368\) \(3.3110\)  
176400.ch2 176400dv5 \([0, 0, 0, -23817675, 44716824250]\) \(272223782641/164025\) \(900338431809600000000\) \([2, 2]\) \(9437184\) \(2.9644\)  
176400.ch3 176400dv8 \([0, 0, 0, -19407675, 61787934250]\) \(-147281603041/215233605\) \(-1181424090220557120000000\) \([2]\) \(18874368\) \(3.3110\)  
176400.ch4 176400dv4 \([0, 0, 0, -14115675, -20412701750]\) \(56667352321/15\) \(82335476160000000\) \([2]\) \(4718592\) \(2.6179\)  
176400.ch5 176400dv3 \([0, 0, 0, -1767675, 418374250]\) \(111284641/50625\) \(277882232040000000000\) \([2, 2]\) \(4718592\) \(2.6179\)  
176400.ch6 176400dv2 \([0, 0, 0, -885675, -316331750]\) \(13997521/225\) \(1235032142400000000\) \([2, 2]\) \(2359296\) \(2.2713\)  
176400.ch7 176400dv1 \([0, 0, 0, -3675, -13805750]\) \(-1/15\) \(-82335476160000000\) \([2]\) \(1179648\) \(1.9247\) \(\Gamma_0(N)\)-optimal
176400.ch8 176400dv6 \([0, 0, 0, 6170325, 3141108250]\) \(4733169839/3515625\) \(-19297377225000000000000\) \([2]\) \(9437184\) \(2.9644\)