# Properties

 Label 190575ed Number of curves $6$ Conductor $190575$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("190575.dh1")

sage: E.isogeny_class()

## Elliptic curves in class 190575ed

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
190575.dh4 190575ed1 [1, -1, 0, -7215192, 7461440091] [2] 5898240 $$\Gamma_0(N)$$-optimal
190575.dh3 190575ed2 [1, -1, 0, -7351317, 7165368216] [2, 2] 11796480
190575.dh5 190575ed3 [1, -1, 0, 6941808, 31649491341] [2] 23592960
190575.dh2 190575ed4 [1, -1, 0, -23822442, -36268988409] [2, 2] 23592960
190575.dh6 190575ed5 [1, -1, 0, 49548933, -215662000284] [2] 47185920
190575.dh1 190575ed6 [1, -1, 0, -360731817, -2636872454034] [2] 47185920

## Rank

sage: E.rank()

The elliptic curves in class 190575ed have rank $$1$$.

## Modular form 190575.2.a.dh

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.