# Properties

 Label 102.a Number of curves $2$ Conductor $102$ CM no Rank $1$ Graph # Related objects

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

E.isogeny_class()

## Elliptic curves in class 102.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
102.a1 102a1 $$[1, 1, 0, -2, 0]$$ $$1771561/612$$ $$612$$ $$$$ $$8$$ $$-0.76353$$ $$\Gamma_0(N)$$-optimal
102.a2 102a2 $$[1, 1, 0, 8, 10]$$ $$46268279/46818$$ $$-46818$$ $$$$ $$16$$ $$-0.41695$$

## Rank

sage: E.rank()

The elliptic curves in class 102.a have rank $$1$$.

## Complex multiplication

The elliptic curves in class 102.a do not have complex multiplication.

## Modular form102.2.a.a

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} - 2 q^{7} - q^{8} + q^{9} + 4 q^{10} - q^{12} - 6 q^{13} + 2 q^{14} + 4 q^{15} + q^{16} - q^{17} - q^{18} + 4 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 