# Properties

 Label 6045.f Number of curves 3 Conductor 6045 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6045.f1")

sage: E.isogeny_class()

## Elliptic curves in class 6045.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6045.f1 6045h1 [0, 1, 1, -3773731, 2820401431]  58320 $$\Gamma_0(N)$$-optimal
6045.f2 6045h2 [0, 1, 1, -3758071, 2844985660]  174960
6045.f3 6045h3 [0, 1, 1, 14606639, 13595424541] [] 524880

## Rank

sage: E.rank()

The elliptic curves in class 6045.f have rank $$0$$.

## Modular form6045.2.a.f

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} - q^{5} - q^{7} + q^{9} - 3q^{11} - 2q^{12} + q^{13} - q^{15} + 4q^{16} - 3q^{17} + 2q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 