# Properties

 Label 474320.bj Number of curves 4 Conductor 474320 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("474320.bj1")

sage: E.isogeny_class()

## Elliptic curves in class 474320.bj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
474320.bj1 474320bj4 [0, 1, 0, -2477470200, 46117850491348] [u'2'] 477757440 $$\Gamma_0(N)$$-optimal*
474320.bj2 474320bj2 [0, 1, 0, -339235640, -2383668253100] [u'2'] 159252480 $$\Gamma_0(N)$$-optimal*
474320.bj3 474320bj1 [0, 1, 0, -5314360, -91766155692] [u'2'] 79626240 $$\Gamma_0(N)$$-optimal*
474320.bj4 474320bj3 [0, 1, 0, 47809480, 2471926614100] [u'2'] 238878720 $$\Gamma_0(N)$$-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 474320.bj3.

## Rank

sage: E.rank()

The elliptic curves in class 474320.bj have rank $$1$$.

## Modular form 474320.2.a.bj

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{5} + q^{9} - 4q^{13} - 2q^{15} + 4q^{19} + O(q^{20})$$

## 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. 