Properties

Label 8400c
Number of curves $6$
Conductor $8400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 8400c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8400.l4 8400c1 \([0, -1, 0, -4383, -110238]\) \(37256083456/525\) \(131250000\) \([2]\) \(6144\) \(0.69791\) \(\Gamma_0(N)\)-optimal
8400.l3 8400c2 \([0, -1, 0, -4508, -103488]\) \(2533446736/275625\) \(1102500000000\) \([2, 2]\) \(12288\) \(1.0445\)  
8400.l2 8400c3 \([0, -1, 0, -17008, 746512]\) \(34008619684/4862025\) \(77792400000000\) \([2, 2]\) \(24576\) \(1.3911\)  
8400.l5 8400c4 \([0, -1, 0, 5992, -523488]\) \(1486779836/8203125\) \(-131250000000000\) \([2]\) \(24576\) \(1.3911\)  
8400.l1 8400c5 \([0, -1, 0, -262008, 51706512]\) \(62161150998242/1607445\) \(51438240000000\) \([2]\) \(49152\) \(1.7376\)  
8400.l6 8400c6 \([0, -1, 0, 27992, 3986512]\) \(75798394558/259416045\) \(-8301313440000000\) \([2]\) \(49152\) \(1.7376\)  

Rank

sage: E.rank()
 

The elliptic curves in class 8400c have rank \(1\).

Complex multiplication

The elliptic curves in class 8400c do not have complex multiplication.

Modular form 8400.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{7} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.