# Properties

 Label 4598.i Number of curves $2$ Conductor $4598$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 4598.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4598.i1 4598a2 $$[1, 1, 0, -28921, -1903095]$$ $$1134626507/1444$$ $$3404876465804$$ $$$$ $$23232$$ $$1.3128$$
4598.i2 4598a1 $$[1, 1, 0, -2301, -13075]$$ $$571787/304$$ $$716816098064$$ $$$$ $$11616$$ $$0.96624$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4598.i have rank $$0$$.

## Complex multiplication

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

## Modular form4598.2.a.i

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 