# Properties

 Label 192d Number of curves $6$ Conductor $192$ CM no Rank $0$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 192d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
192.b5 192d1 $$[0, -1, 0, 3, -3]$$ $$2048/3$$ $$-3072$$ $$$$ $$8$$ $$-0.64535$$ $$\Gamma_0(N)$$-optimal
192.b4 192d2 $$[0, -1, 0, -17, -15]$$ $$35152/9$$ $$147456$$ $$[2, 2]$$ $$16$$ $$-0.29878$$
192.b2 192d3 $$[0, -1, 0, -257, -1503]$$ $$28756228/3$$ $$196608$$ $$$$ $$32$$ $$0.047795$$
192.b3 192d4 $$[0, -1, 0, -97, 385]$$ $$1556068/81$$ $$5308416$$ $$[2, 2]$$ $$32$$ $$0.047795$$
192.b1 192d5 $$[0, -1, 0, -1537, 23713]$$ $$3065617154/9$$ $$1179648$$ $$$$ $$64$$ $$0.39437$$
192.b6 192d6 $$[0, -1, 0, 63, 1377]$$ $$207646/6561$$ $$-859963392$$ $$$$ $$64$$ $$0.39437$$

## Rank

sage: E.rank()

The elliptic curves in class 192d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 192d do not have complex multiplication.

## Modular form192.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{9} + 4q^{11} + 2q^{13} - 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 