# Properties

 Label 31200.bt Number of curves $2$ Conductor $31200$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bt1")

sage: E.isogeny_class()

## Elliptic curves in class 31200.bt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31200.bt1 31200bx2 $$[0, 1, 0, -32833, -2299537]$$ $$61162984000/41067$$ $$2628288000000$$ $$$$ $$92160$$ $$1.3215$$
31200.bt2 31200bx1 $$[0, 1, 0, -2458, -21412]$$ $$1643032000/767637$$ $$767637000000$$ $$$$ $$46080$$ $$0.97496$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 31200.bt have rank $$1$$.

## Complex multiplication

The elliptic curves in class 31200.bt do not have complex multiplication.

## Modular form 31200.2.a.bt

sage: E.q_eigenform(10)

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