# Properties

 Label 16744d Number of curves $4$ Conductor $16744$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 16744d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16744.f4 16744d1 $$[0, 0, 0, -146, 1185]$$ $$-21511084032/25465531$$ $$-407448496$$ $$$$ $$4096$$ $$0.34605$$ $$\Gamma_0(N)$$-optimal
16744.f3 16744d2 $$[0, 0, 0, -2791, 56730]$$ $$9392111857872/4380649$$ $$1121446144$$ $$[2, 2]$$ $$8192$$ $$0.69262$$
16744.f2 16744d3 $$[0, 0, 0, -3251, 36766]$$ $$3710860803108/1577224103$$ $$1615077481472$$ $$$$ $$16384$$ $$1.0392$$
16744.f1 16744d4 $$[0, 0, 0, -44651, 3631574]$$ $$9614292367656708/2093$$ $$2143232$$ $$$$ $$16384$$ $$1.0392$$

## Rank

sage: E.rank()

The elliptic curves in class 16744d have rank $$1$$.

## Complex multiplication

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

## Modular form 16744.2.a.d

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} - 3q^{9} + 4q^{11} + q^{13} - 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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 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. 