# Properties

 Label 1386.k Number of curves $4$ Conductor $1386$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 1386.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1386.k1 1386l4 $$[1, -1, 1, -2039, 35925]$$ $$1285429208617/614922$$ $$448278138$$ $$$$ $$1024$$ $$0.61497$$
1386.k2 1386l3 $$[1, -1, 1, -1139, -14259]$$ $$223980311017/4278582$$ $$3119086278$$ $$$$ $$1024$$ $$0.61497$$
1386.k3 1386l2 $$[1, -1, 1, -149, 393]$$ $$498677257/213444$$ $$155600676$$ $$[2, 2]$$ $$512$$ $$0.26839$$
1386.k4 1386l1 $$[1, -1, 1, 31, 33]$$ $$4657463/3696$$ $$-2694384$$ $$$$ $$256$$ $$-0.078180$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1386.k have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1386.k do not have complex multiplication.

## Modular form1386.2.a.k

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 2 q^{5} + q^{7} + q^{8} + 2 q^{10} - q^{11} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})$$ ## 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. 