# Properties

 Label 3528k Number of curves $4$ Conductor $3528$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 3528k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3528.x4 3528k1 [0, 0, 0, 441, 18522]  3072 $$\Gamma_0(N)$$-optimal
3528.x3 3528k2 [0, 0, 0, -8379, 277830] [2, 2] 6144
3528.x2 3528k3 [0, 0, 0, -26019, -1278018]  12288
3528.x1 3528k4 [0, 0, 0, -131859, 18429390]  12288

## Rank

sage: E.rank()

The elliptic curves in class 3528k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3528k do not have complex multiplication.

## Modular form3528.2.a.k

sage: E.q_eigenform(10)

$$q + 2q^{5} + 4q^{11} - 2q^{13} - 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 