# Properties

 Label 46410n Number of curves 4 Conductor 46410 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("46410.p1")

sage: E.isogeny_class()

## Elliptic curves in class 46410n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
46410.p4 46410n1 [1, 1, 0, -332, -160944]  122880 $$\Gamma_0(N)$$-optimal
46410.p3 46410n2 [1, 1, 0, -46412, -3819696] [2, 2] 245760
46410.p2 46410n3 [1, 1, 0, -90092, 4453296]  491520
46410.p1 46410n4 [1, 1, 0, -740012, -245331216]  491520

## Rank

sage: E.rank()

The elliptic curves in class 46410n have rank $$1$$.

## Modular form 46410.2.a.p

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 