# Properties

 Label 45675.e Number of curves $6$ Conductor $45675$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("45675.e1")

sage: E.isogeny_class()

## Elliptic curves in class 45675.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
45675.e1 45675k4 [1, -1, 1, -46040405, -120230661778] [2] 1572864
45675.e2 45675k6 [1, -1, 1, -9555530, 9244638722] [2] 3145728
45675.e3 45675k3 [1, -1, 1, -2932655, -1802316778] [2, 2] 1572864
45675.e4 45675k2 [1, -1, 1, -2877530, -1878058528] [2, 2] 786432
45675.e5 45675k1 [1, -1, 1, -176405, -30489028] [2] 393216 $$\Gamma_0(N)$$-optimal
45675.e6 45675k5 [1, -1, 1, 2808220, -8002461778] [2] 3145728

## Rank

sage: E.rank()

The elliptic curves in class 45675.e have rank $$0$$.

## Modular form 45675.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - q^{7} + 3q^{8} - 4q^{11} + 2q^{13} + q^{14} - q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.