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Properties

Label 327600bi
Number of curves $4$
Conductor $327600$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bi1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 327600bi have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 327600bi do not have complex multiplication.

Modular form 327600.2.a.bi

Copy content sage:E.q_eigenform(10)
 
\(q - q^{7} - 4 q^{11} + q^{13} - 2 q^{17} - 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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 sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.

Elliptic curves in class 327600bi

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
327600.bi3 327600bi1 \([0, 0, 0, -500004516675, 136084500904509250]\) \(296304326013275547793071733369/268420373544960000000\) \(12523420948113653760000000000000\) \([2]\) \(1981808640\) \(5.2653\) \(\Gamma_0(N)\)-optimal
327600.bi2 327600bi2 \([0, 0, 0, -503617188675, 134018186189373250]\) \(302773487204995438715379645049/8911747415025000000000000\) \(415786487395406400000000000000000000\) \([2, 2]\) \(3963617280\) \(5.6119\)  
327600.bi4 327600bi3 \([0, 0, 0, 126382811325, 446938556189373250]\) \(4784981304203817469820354951/1852343836482910078035000000\) \(-86422954034946652600800960000000000000\) \([2]\) \(7927234560\) \(5.9585\)  
327600.bi1 327600bi4 \([0, 0, 0, -1191419940675, -311146325579330750]\) \(4008766897254067912673785886329/1423480510711669921875000000\) \(66413906707763671875000000000000000000\) \([2]\) \(7927234560\) \(5.9585\)