# Properties

 Label 16744.e Number of curves $4$ Conductor $16744$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 16744.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16744.e1 16744g3 $$[0, 0, 0, -15851, -227386]$$ $$215062038362754/113550802729$$ $$232552043988992$$ $$$$ $$35840$$ $$1.4482$$
16744.e2 16744g2 $$[0, 0, 0, -9091, 330990]$$ $$81144432781668/740329681$$ $$758097593344$$ $$[2, 2]$$ $$17920$$ $$1.1017$$
16744.e3 16744g1 $$[0, 0, 0, -9071, 332530]$$ $$322440248841552/27209$$ $$6965504$$ $$$$ $$8960$$ $$0.75509$$ $$\Gamma_0(N)$$-optimal
16744.e4 16744g4 $$[0, 0, 0, -2651, 790806]$$ $$-1006057824354/131332646081$$ $$-268969259173888$$ $$$$ $$35840$$ $$1.4482$$

## Rank

sage: E.rank()

The elliptic curves in class 16744.e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 16744.e do not have complex multiplication.

## Modular form 16744.2.a.e

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} - 3q^{9} - q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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. 