# Properties

 Label 3850.j Number of curves $4$ Conductor $3850$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 3850.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3850.j1 3850c4 $$[1, 1, 0, -88000, -10029750]$$ $$4823468134087681/30382271150$$ $$474722986718750$$ $$$$ $$27648$$ $$1.6535$$
3850.j2 3850c2 $$[1, 1, 0, -6750, 201500]$$ $$2177286259681/105875000$$ $$1654296875000$$ $$$$ $$9216$$ $$1.1042$$
3850.j3 3850c3 $$[1, 1, 0, -2250, -340000]$$ $$-80677568161/3131816380$$ $$-48934630937500$$ $$$$ $$13824$$ $$1.3070$$
3850.j4 3850c1 $$[1, 1, 0, 250, 12500]$$ $$109902239/4312000$$ $$-67375000000$$ $$$$ $$4608$$ $$0.75766$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3850.j have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3850.j do not have complex multiplication.

## Modular form3850.2.a.j

sage: E.q_eigenform(10)

$$q - q^{2} + 2q^{3} + q^{4} - 2q^{6} - q^{7} - q^{8} + q^{9} - q^{11} + 2q^{12} - 2q^{13} + q^{14} + q^{16} - 6q^{17} - q^{18} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 