Properties

Label 4998g
Number of curves $4$
Conductor $4998$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 4998g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4998.d2 4998g1 \([1, 1, 0, -12520, -544256]\) \(1845026709625/793152\) \(93313539648\) \([2]\) \(8640\) \(1.0651\) \(\Gamma_0(N)\)-optimal
4998.d3 4998g2 \([1, 1, 0, -10560, -717912]\) \(-1107111813625/1228691592\) \(-144554337107208\) \([2]\) \(17280\) \(1.4116\)  
4998.d1 4998g3 \([1, 1, 0, -36775, 2036917]\) \(46753267515625/11591221248\) \(1363695588605952\) \([2]\) \(25920\) \(1.6144\)  
4998.d4 4998g4 \([1, 1, 0, 88665, 13050549]\) \(655215969476375/1001033261568\) \(-117770562190213632\) \([2]\) \(51840\) \(1.9610\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4998g have rank \(1\).

Complex multiplication

The elliptic curves in class 4998g do not have complex multiplication.

Modular form 4998.2.a.g

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{12} - 2 q^{13} + q^{16} + q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.