# Properties

 Label 88305k Number of curves 4 Conductor 88305 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("88305.k1")

sage: E.isogeny_class()

## Elliptic curves in class 88305k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
88305.k3 88305k1 [1, 1, 1, -2120, 34712]  100352 $$\Gamma_0(N)$$-optimal
88305.k2 88305k2 [1, 1, 1, -6325, -151990] [2, 2] 200704
88305.k4 88305k3 [1, 1, 1, 14700, -925710]  401408
88305.k1 88305k4 [1, 1, 1, -94630, -11243098]  401408

## Rank

sage: E.rank()

The elliptic curves in class 88305k have rank $$1$$.

## Modular form 88305.2.a.k

sage: E.q_eigenform(10)

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

## 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. 