# Properties

 Label 17745n Number of curves 4 Conductor 17745 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("17745.f1")

sage: E.isogeny_class()

## Elliptic curves in class 17745n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
17745.f3 17745n1 [1, 0, 0, -426, 3171]  7680 $$\Gamma_0(N)$$-optimal
17745.f2 17745n2 [1, 0, 0, -1271, -13560] [2, 2] 15360
17745.f1 17745n3 [1, 0, 0, -19016, -1010829]  30720
17745.f4 17745n4 [1, 0, 0, 2954, -83695]  30720

## Rank

sage: E.rank()

The elliptic curves in class 17745n have rank $$0$$.

## Modular form 17745.2.a.f

sage: E.q_eigenform(10)

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

## 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. 