# Properties

 Label 3192.m Number of curves $4$ Conductor $3192$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("m1")

E.isogeny_class()

## Elliptic curves in class 3192.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3192.m1 3192i4 $$[0, 1, 0, -2864, 58032]$$ $$2538016415428/872613$$ $$893555712$$ $$[2]$$ $$2048$$ $$0.68876$$
3192.m2 3192i3 $$[0, 1, 0, -1464, -21600]$$ $$339112345828/8210223$$ $$8407268352$$ $$[2]$$ $$2048$$ $$0.68876$$
3192.m3 3192i2 $$[0, 1, 0, -204, 576]$$ $$3685542352/1432809$$ $$366799104$$ $$[2, 2]$$ $$1024$$ $$0.34218$$
3192.m4 3192i1 $$[0, 1, 0, 41, 86]$$ $$464857088/410571$$ $$-6569136$$ $$[4]$$ $$512$$ $$-0.0043898$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3192.2.a.m

sage: E.q_eigenform(10)

$$q + q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} - q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.