Properties

Label 2200.e
Number of curves $4$
Conductor $2200$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 2200.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2200.e1 2200f3 \([0, 0, 0, -188675, -1623250]\) \(46424454082884/26794860125\) \(428717762000000000\) \([2]\) \(27648\) \(2.0728\)  
2200.e2 2200f2 \([0, 0, 0, -126175, 17189250]\) \(55537159171536/228765625\) \(915062500000000\) \([2, 2]\) \(13824\) \(1.7263\)  
2200.e3 2200f1 \([0, 0, 0, -126050, 17225125]\) \(885956203616256/15125\) \(3781250000\) \([4]\) \(6912\) \(1.3797\) \(\Gamma_0(N)\)-optimal
2200.e4 2200f4 \([0, 0, 0, -65675, 33705750]\) \(-1957960715364/29541015625\) \(-472656250000000000\) \([2]\) \(27648\) \(2.0728\)  

Rank

sage: E.rank()
 

The elliptic curves in class 2200.e have rank \(0\).

Complex multiplication

The elliptic curves in class 2200.e do not have complex multiplication.

Modular form 2200.2.a.e

sage: E.q_eigenform(10)
 
\(q - 4 q^{7} - 3 q^{9} - q^{11} - 6 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with LMFDB labels.