# Properties

 Label 39039h Number of curves 6 Conductor 39039 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("39039.r1")

sage: E.isogeny_class()

## Elliptic curves in class 39039h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
39039.r4 39039h1 [1, 1, 0, -5749, 165352] [2] 38400 $$\Gamma_0(N)$$-optimal
39039.r3 39039h2 [1, 1, 0, -6594, 112455] [2, 2] 76800
39039.r6 39039h3 [1, 1, 0, 21291, 843042] [2] 153600
39039.r2 39039h4 [1, 1, 0, -47999, -3986640] [2, 2] 153600
39039.r5 39039h5 [1, 1, 0, 5236, -12280653] [2] 307200
39039.r1 39039h6 [1, 1, 0, -763714, -257206607] [2] 307200

## Rank

sage: E.rank()

The elliptic curves in class 39039h have rank $$0$$.

## Modular form 39039.2.a.r

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + 2q^{5} - q^{6} - q^{7} - 3q^{8} + q^{9} + 2q^{10} + q^{11} + q^{12} - q^{14} - 2q^{15} - q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.