# Properties

 Label 1386a Number of curves $2$ Conductor $1386$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 1386a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1386.d1 1386a1 $$[1, -1, 0, -231, -451]$$ $$69426531/34496$$ $$678984768$$ $$$$ $$576$$ $$0.38764$$ $$\Gamma_0(N)$$-optimal
1386.d2 1386a2 $$[1, -1, 0, 849, -4123]$$ $$3436115229/2324168$$ $$-45746598744$$ $$$$ $$1152$$ $$0.73421$$

## Rank

sage: E.rank()

The elliptic curves in class 1386a have rank $$1$$.

## Complex multiplication

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

## Modular form1386.2.a.a

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels. 