# Properties

 Label 46389k Number of curves 6 Conductor 46389 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("46389.d1")

sage: E.isogeny_class()

## Elliptic curves in class 46389k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
46389.d6 46389k1 [1, 0, 0, 2163, 8712] [2] 52992 $$\Gamma_0(N)$$-optimal
46389.d5 46389k2 [1, 0, 0, -8882, 68355] [2, 2] 105984
46389.d3 46389k3 [1, 0, 0, -86197, -9688798] [2] 211968
46389.d2 46389k4 [1, 0, 0, -108287, 13686840] [2, 2] 211968
46389.d4 46389k5 [1, 0, 0, -75152, 22229043] [2] 423936
46389.d1 46389k6 [1, 0, 0, -1731902, 877125297] [2] 423936

## Rank

sage: E.rank()

The elliptic curves in class 46389k have rank $$1$$.

## Modular form 46389.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} + 2q^{13} + q^{14} + 2q^{15} - q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.