# Properties

 Label 3675.f Number of curves 4 Conductor 3675 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("3675.f1")

sage: E.isogeny_class()

## Elliptic curves in class 3675.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3675.f1 3675l3 [1, 0, 0, -137838, 19684167]  18432
3675.f2 3675l2 [1, 0, 0, -9213, 261792] [2, 2] 9216
3675.f3 3675l1 [1, 0, 0, -3088, -62833]  4608 $$\Gamma_0(N)$$-optimal
3675.f4 3675l4 [1, 0, 0, 21412, 1639917]  18432

## Rank

sage: E.rank()

The elliptic curves in class 3675.f have rank $$1$$.

## Modular form3675.2.a.f

sage: E.q_eigenform(10)

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

## 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 LMFDB 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 LMFDB labels. 