# Properties

 Label 3450.d Number of curves $6$ Conductor $3450$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 3450.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
3450.d1 3450d4 [1, 1, 0, -2760000, 1763716500] [2] 36864
3450.d2 3450d5 [1, 1, 0, -645750, -171356250] [2] 73728
3450.d3 3450d3 [1, 1, 0, -177000, 25987500] [2, 2] 36864
3450.d4 3450d2 [1, 1, 0, -172500, 27504000] [2, 2] 18432
3450.d5 3450d1 [1, 1, 0, -10500, 450000] [2] 9216 $$\Gamma_0(N)$$-optimal
3450.d6 3450d6 [1, 1, 0, 219750, 126365250] [2] 73728

## Rank

sage: E.rank()

The elliptic curves in class 3450.d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 3450.d do not have complex multiplication.

## Modular form3450.2.a.d

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.