# Properties

 Label 55470.x Number of curves $4$ Conductor $55470$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 55470.x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55470.x1 55470bd4 $$[1, 1, 1, -1550425, -738640015]$$ $$65202655558249/512820150$$ $$3241722346992637350$$ $$$$ $$1419264$$ $$2.3801$$
55470.x2 55470bd2 $$[1, 1, 1, -163675, 6322085]$$ $$76711450249/41602500$$ $$262984506246022500$$ $$[2, 2]$$ $$709632$$ $$2.0335$$
55470.x3 55470bd1 $$[1, 1, 1, -126695, 17282957]$$ $$35578826569/51600$$ $$326182333328400$$ $$$$ $$354816$$ $$1.6869$$ $$\Gamma_0(N)$$-optimal
55470.x4 55470bd3 $$[1, 1, 1, 631395, 50527977]$$ $$4403686064471/2721093750$$ $$-17201021484114843750$$ $$$$ $$1419264$$ $$2.3801$$

## Rank

sage: E.rank()

The elliptic curves in class 55470.x have rank $$0$$.

## Complex multiplication

The elliptic curves in class 55470.x do not have complex multiplication.

## Modular form 55470.2.a.x

sage: E.q_eigenform(10)

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