# Properties

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

# Related objects

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 form4998.2.a.g

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - q^{12} - 2 q^{13} + q^{16} + q^{17} - q^{18} + 4 q^{19} + O(q^{20})$$

## 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.