# Properties

 Label 436800.hy Number of curves 8 Conductor 436800 CM no Rank 2 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("436800.hy1")

sage: E.isogeny_class()

## Elliptic curves in class 436800.hy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
436800.hy1 436800hy7 [0, -1, 0, -64304728033, 6276448294455937] [u'2'] 509607936 $$\Gamma_0(N)$$-optimal*
436800.hy2 436800hy6 [0, -1, 0, -4019048033, 98070379655937] [u'2', u'2'] 254803968 $$\Gamma_0(N)$$-optimal*
436800.hy3 436800hy8 [0, -1, 0, -3969656033, 100598212823937] [u'2'] 509607936
436800.hy4 436800hy4 [0, -1, 0, -794248033, 8601647655937] [u'2'] 169869312 $$\Gamma_0(N)$$-optimal*
436800.hy5 436800hy3 [0, -1, 0, -254280033, 1492786151937] [u'2'] 127401984 $$\Gamma_0(N)$$-optimal*
436800.hy6 436800hy2 [0, -1, 0, -66248033, 36727655937] [u'2', u'2'] 84934656 $$\Gamma_0(N)$$-optimal*
436800.hy7 436800hy1 [0, -1, 0, -41160033, -101080728063] [u'2'] 42467328 $$\Gamma_0(N)$$-optimal*
436800.hy8 436800hy5 [0, -1, 0, 260343967, 291142823937] [u'2'] 169869312
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 6 curves highlighted, and conditionally curve 436800.hy7.

## Rank

sage: E.rank()

The elliptic curves in class 436800.hy have rank $$2$$.

## Modular form 436800.2.a.hy

sage: E.q_eigenform(10)

$$q - q^{3} + q^{7} + q^{9} + q^{13} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.