# Properties

 Label 336e Number of curves $6$ Conductor $336$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 336e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
336.a6 336e1 $$[0, -1, 0, 16, 0]$$ $$103823/63$$ $$-258048$$ $$[2]$$ $$32$$ $$-0.27237$$ $$\Gamma_0(N)$$-optimal
336.a5 336e2 $$[0, -1, 0, -64, 64]$$ $$7189057/3969$$ $$16257024$$ $$[2, 2]$$ $$64$$ $$0.074205$$
336.a3 336e3 $$[0, -1, 0, -624, -5760]$$ $$6570725617/45927$$ $$188116992$$ $$[2]$$ $$128$$ $$0.42078$$
336.a2 336e4 $$[0, -1, 0, -784, 8704]$$ $$13027640977/21609$$ $$88510464$$ $$[2, 4]$$ $$128$$ $$0.42078$$
336.a1 336e5 $$[0, -1, 0, -12544, 544960]$$ $$53297461115137/147$$ $$602112$$ $$[4]$$ $$256$$ $$0.76735$$
336.a4 336e6 $$[0, -1, 0, -544, 13888]$$ $$-4354703137/17294403$$ $$-70837874688$$ $$[4]$$ $$256$$ $$0.76735$$

## Rank

sage: E.rank()

The elliptic curves in class 336e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 336e do not have complex multiplication.

## Modular form336.2.a.e

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{5} + q^{7} + q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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 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.