# Properties

 Label 90354.k Number of curves 4 Conductor 90354 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("90354.k1")

sage: E.isogeny_class()

## Elliptic curves in class 90354.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
90354.k1 90354k3 [1, 0, 1, -13779014, -19687955836]  5184000
90354.k2 90354k4 [1, 0, 1, -13765324, -19729025836]  10368000
90354.k3 90354k1 [1, 0, 1, -61634, 4287764]  1036800 $$\Gamma_0(N)$$-optimal
90354.k4 90354k2 [1, 0, 1, 157406, 27944084]  2073600

## Rank

sage: E.rank()

The elliptic curves in class 90354.k have rank $$1$$.

## Modular form 90354.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + 4q^{5} - q^{6} - 2q^{7} - q^{8} + q^{9} - 4q^{10} + q^{11} + q^{12} - 4q^{13} + 2q^{14} + 4q^{15} + q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 