↑

Properties

Label 58800fc
Number of curves $8$
Conductor $58800$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("fc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 58800fc have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 58800fc do not have complex multiplication.

Modular form 58800.2.a.fc

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 58800fc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58800.cu8 58800fc1 \([0, -1, 0, 28992, 5824512]\) \(357911/2160\) \(-16263797760000000\) \([2]\) \(331776\) \(1.7925\) \(\Gamma_0(N)\)-optimal
58800.cu6 58800fc2 \([0, -1, 0, -363008, 76384512]\) \(702595369/72900\) \(548903174400000000\) \([2, 2]\) \(663552\) \(2.1391\)  
58800.cu7 58800fc3 \([0, -1, 0, -265008, -171751488]\) \(-273359449/1536000\) \(-11565367296000000000\) \([2]\) \(995328\) \(2.3418\)  
58800.cu5 58800fc4 \([0, -1, 0, -1343008, -515535488]\) \(35578826569/5314410\) \(40015041413760000000\) \([2]\) \(1327104\) \(2.4857\)  
58800.cu4 58800fc5 \([0, -1, 0, -5655008, 5177872512]\) \(2656166199049/33750\) \(254121840000000000\) \([2]\) \(1327104\) \(2.4857\)  
58800.cu3 58800fc6 \([0, -1, 0, -6537008, -6418663488]\) \(4102915888729/9000000\) \(67765824000000000000\) \([2, 2]\) \(1990656\) \(2.6884\)  
58800.cu1 58800fc7 \([0, -1, 0, -104537008, -411354663488]\) \(16778985534208729/81000\) \(609892416000000000\) \([2]\) \(3981312\) \(3.0350\)  
58800.cu2 58800fc8 \([0, -1, 0, -8889008, -1385383488]\) \(10316097499609/5859375000\) \(44118375000000000000000\) \([2]\) \(3981312\) \(3.0350\)