# Properties

 Label 3675.d Number of curves $2$ Conductor $3675$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 3675.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.d1 3675h1 $$[1, 1, 1, -883, -10144]$$ $$5177717/189$$ $$2779457625$$ $$$$ $$2304$$ $$0.58174$$ $$\Gamma_0(N)$$-optimal
3675.d2 3675h2 $$[1, 1, 1, 342, -34644]$$ $$300763/35721$$ $$-525317491125$$ $$$$ $$4608$$ $$0.92831$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3675.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{6} + 3q^{8} + q^{9} - 6q^{11} + q^{12} + 2q^{13} - q^{16} - 4q^{17} - q^{18} + 6q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 