# Properties

 Label 407330.u Number of curves 4 Conductor 407330 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("407330.u1")

sage: E.isogeny_class()

## Elliptic curves in class 407330.u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
407330.u1 407330u4 [1, 0, 0, -13815375, 19203578125] [u'2'] 43794432 $$\Gamma_0(N)$$-optimal*
407330.u2 407330u2 [1, 0, 0, -1891715, -992717183] [u'2'] 14598144 $$\Gamma_0(N)$$-optimal*
407330.u3 407330u1 [1, 0, 0, -29635, -38214975] [u'2'] 7299072 $$\Gamma_0(N)$$-optimal*
407330.u4 407330u3 [1, 0, 0, 266605, 1029374737] [u'2'] 21897216 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 407330.u3.

## Rank

sage: E.rank()

The elliptic curves in class 407330.u have rank $$1$$.

## Modular form 407330.2.a.u

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 