# Properties

 Label 36414bh Number of curves $4$ Conductor $36414$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bh1")

sage: E.isogeny_class()

## Elliptic curves in class 36414bh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
36414.l4 36414bh1 $$[1, -1, 0, 2547, -13367835]$$ $$103823/4386816$$ $$-77191676866031616$$ $$$$ $$442368$$ $$1.9193$$ $$\Gamma_0(N)$$-optimal
36414.l3 36414bh2 $$[1, -1, 0, -829773, -285536475]$$ $$3590714269297/73410624$$ $$1291754467554997824$$ $$[2, 2]$$ $$884736$$ $$2.2659$$
36414.l2 36414bh3 $$[1, -1, 0, -1766133, 477971469]$$ $$34623662831857/14438442312$$ $$254062986320087835912$$ $$$$ $$1769472$$ $$2.6125$$
36414.l1 36414bh4 $$[1, -1, 0, -13210533, -18477825219]$$ $$14489843500598257/6246072$$ $$109907680537767672$$ $$$$ $$1769472$$ $$2.6125$$

## Rank

sage: E.rank()

The elliptic curves in class 36414bh have rank $$1$$.

## Complex multiplication

The elliptic curves in class 36414bh do not have complex multiplication.

## Modular form 36414.2.a.bh

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2q^{5} + q^{7} - q^{8} + 2q^{10} - 6q^{13} - q^{14} + q^{16} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 