# Properties

 Label 3192.d Number of curves $4$ Conductor $3192$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 3192.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3192.d1 3192n3 $$[0, -1, 0, -17024, -849300]$$ $$266442869452034/399$$ $$817152$$ $$$$ $$3072$$ $$0.83235$$
3192.d2 3192n2 $$[0, -1, 0, -1064, -12996]$$ $$130213720228/159201$$ $$163021824$$ $$[2, 2]$$ $$1536$$ $$0.48578$$
3192.d3 3192n4 $$[0, -1, 0, -784, -20276]$$ $$-26055281954/73892007$$ $$-151330830336$$ $$$$ $$3072$$ $$0.83235$$
3192.d4 3192n1 $$[0, -1, 0, -84, -60]$$ $$259108432/136857$$ $$35035392$$ $$$$ $$768$$ $$0.13921$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3192.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3192.d do not have complex multiplication.

## Modular form3192.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 2 q^{13} + 2 q^{15} + 6 q^{17} - q^{19} + O(q^{20})$$

## 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. 