# Properties

 Label 34a Number of curves 4 Conductor 34 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("34.a1")

sage: E.isogeny_class()

## Elliptic curves in class 34a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
34.a4 34a1 [1, 0, 0, -3, 1]  2 $$\Gamma_0(N)$$-optimal
34.a3 34a2 [1, 0, 0, -43, 105]  4
34.a2 34a3 [1, 0, 0, -103, -411]  6
34.a1 34a4 [1, 0, 0, -113, -329]  12

## Rank

sage: E.rank()

The elliptic curves in class 34a have rank $$0$$.

## Modular form34.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 