# Properties

 Label 3675.a Number of curves $2$ Conductor $3675$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 3675.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3675.a1 3675d2 $$[0, -1, 1, -1117608, -454753132]$$ $$-1713910976512/1594323$$ $$-143608669136296875$$ $$[]$$ $$69888$$ $$2.2151$$
3675.a2 3675d1 $$[0, -1, 1, -2858, 64868]$$ $$-28672/3$$ $$-270225046875$$ $$[]$$ $$5376$$ $$0.93259$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form3675.2.a.a

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.