# Properties

 Label 3675.f Number of curves $4$ Conductor $3675$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 3675.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.f1 3675l3 $$[1, 0, 0, -137838, 19684167]$$ $$157551496201/13125$$ $$24127236328125$$ $$[2]$$ $$18432$$ $$1.6117$$
3675.f2 3675l2 $$[1, 0, 0, -9213, 261792]$$ $$47045881/11025$$ $$20266878515625$$ $$[2, 2]$$ $$9216$$ $$1.2651$$
3675.f3 3675l1 $$[1, 0, 0, -3088, -62833]$$ $$1771561/105$$ $$193017890625$$ $$[2]$$ $$4608$$ $$0.91856$$ $$\Gamma_0(N)$$-optimal
3675.f4 3675l4 $$[1, 0, 0, 21412, 1639917]$$ $$590589719/972405$$ $$-1787538685078125$$ $$[2]$$ $$18432$$ $$1.6117$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 3675.f do not have complex multiplication.

## 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.