# Properties

 Label 4598.a Number of curves $2$ Conductor $4598$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 4598.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4598.a1 4598k2 $$[1, 1, 0, -8472, 328750]$$ $$-37966934881/4952198$$ $$-8773120841078$$ $$[]$$ $$14000$$ $$1.2167$$
4598.a2 4598k1 $$[1, 1, 0, -2, -1580]$$ $$-1/608$$ $$-1077109088$$ $$[]$$ $$2800$$ $$0.41201$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4598.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4598.a do not have complex multiplication.

## Modular form4598.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} - 3 q^{7} - q^{8} - 2 q^{9} + 4 q^{10} - q^{12} + q^{13} + 3 q^{14} + 4 q^{15} + q^{16} - 3 q^{17} + 2 q^{18} + q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 