# Properties

 Label 123.a Number of curves 2 Conductor 123 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("123.a1")

sage: E.isogeny_class()

## Elliptic curves in class 123.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
123.a1 123a1 [0, 1, 1, -10, 10]  20 $$\Gamma_0(N)$$-optimal
123.a2 123a2 [0, 1, 1, 20, -890] [] 100

## Rank

sage: E.rank()

The elliptic curves in class 123.a have rank $$1$$.

## Modular form123.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{2} + q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + q^{9} + 8q^{10} - 3q^{11} + 2q^{12} - 6q^{13} + 4q^{14} - 4q^{15} - 4q^{16} + 3q^{17} - 2q^{18} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 