Properties

Label 277200jz
Number of curves $4$
Conductor $277200$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("jz1")
 
E.isogeny_class()
 

Elliptic curves in class 277200jz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277200.jz3 277200jz1 \([0, 0, 0, -732675, 241213250]\) \(932288503609/779625\) \(36374184000000000\) \([2]\) \(3538944\) \(2.1056\) \(\Gamma_0(N)\)-optimal
277200.jz2 277200jz2 \([0, 0, 0, -894675, 126679250]\) \(1697509118089/833765625\) \(38900169000000000000\) \([2, 2]\) \(7077888\) \(2.4522\)  
277200.jz4 277200jz3 \([0, 0, 0, 3263325, 970753250]\) \(82375335041831/56396484375\) \(-2631234375000000000000\) \([2]\) \(14155776\) \(2.7988\)  
277200.jz1 277200jz4 \([0, 0, 0, -7644675, -8047570750]\) \(1058993490188089/13182390375\) \(615037605336000000000\) \([2]\) \(14155776\) \(2.7988\)  

Rank

sage: E.rank()
 

The elliptic curves in class 277200jz have rank \(0\).

Complex multiplication

The elliptic curves in class 277200jz do not have complex multiplication.

Modular form 277200.2.a.jz

sage: E.q_eigenform(10)
 
\(q + q^{7} - q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with Cremona labels.