# Properties

 Label 10304.bh Number of curves $2$ Conductor $10304$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 10304.bh

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10304.bh1 10304bf2 $$[0, -1, 0, -257, 833]$$ $$57512456/25921$$ $$849379328$$ $$$$ $$4608$$ $$0.40859$$
10304.bh2 10304bf1 $$[0, -1, 0, -217, 1305]$$ $$277167808/161$$ $$659456$$ $$$$ $$2304$$ $$0.062021$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 10304.bh have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10304.bh do not have complex multiplication.

## Modular form 10304.2.a.bh

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 