# Properties

 Label 13328w Number of curves 4 Conductor 13328 CM no Rank 2 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("13328.e1")

sage: E.isogeny_class()

## Elliptic curves in class 13328w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13328.e4 13328w1 [0, 1, 0, -2368, 26676] [2] 13824 $$\Gamma_0(N)$$-optimal
13328.e3 13328w2 [0, 1, 0, -33728, 2372404] [2] 27648
13328.e2 13328w3 [0, 1, 0, -80768, -8860748] [2] 41472
13328.e1 13328w4 [0, 1, 0, -88608, -7045004] [2] 82944

## Rank

sage: E.rank()

The elliptic curves in class 13328w have rank $$2$$.

## Modular form 13328.2.a.e

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{9} - 6q^{11} - 2q^{13} + q^{17} - 4q^{19} + O(q^{20})$$

## 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.