# Properties

 Label 13520a Number of curves 4 Conductor 13520 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("13520.l1")

sage: E.isogeny_class()

## Elliptic curves in class 13520a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13520.l3 13520a1 [0, 0, 0, -338, -2197]  4608 $$\Gamma_0(N)$$-optimal
13520.l2 13520a2 [0, 0, 0, -1183, 13182] [2, 2] 9216
13520.l1 13520a3 [0, 0, 0, -18083, 935922]  18432
13520.l4 13520a4 [0, 0, 0, 2197, 74698]  18432

## Rank

sage: E.rank()

The elliptic curves in class 13520a have rank $$1$$.

## Modular form 13520.2.a.l

sage: E.q_eigenform(10)

$$q - q^{5} - 4q^{7} - 3q^{9} + 4q^{11} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 