# Properties

 Label 3192.p Number of curves $2$ Conductor $3192$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("p1")

E.isogeny_class()

## Elliptic curves in class 3192.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3192.p1 3192g1 $$[0, 1, 0, -4955392, -4246324048]$$ $$13141891860831409148932/4237307541832617$$ $$4339002922836599808$$ $$[2]$$ $$94080$$ $$2.5506$$ $$\Gamma_0(N)$$-optimal
3192.p2 3192g2 $$[0, 1, 0, -4283112, -5439217680]$$ $$-4242991426585187031506/3781894171664380023$$ $$-7745319263568650287104$$ $$[2]$$ $$188160$$ $$2.8972$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3192.2.a.p

sage: E.q_eigenform(10)

$$q + q^{3} + 2 q^{5} - q^{7} + q^{9} - 6 q^{11} - 2 q^{13} + 2 q^{15} + q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.