# Properties

 Label 8400bt Number of curves $2$ Conductor $8400$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 8400bt

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8400.bd1 8400bt1 $$[0, -1, 0, -2728, -53648]$$ $$4386781853/27216$$ $$13934592000$$ $$[2]$$ $$7680$$ $$0.78449$$ $$\Gamma_0(N)$$-optimal
8400.bd2 8400bt2 $$[0, -1, 0, -1128, -117648]$$ $$-310288733/11573604$$ $$-5925685248000$$ $$[2]$$ $$15360$$ $$1.1311$$

## Rank

sage: E.rank()

The elliptic curves in class 8400bt have rank $$0$$.

## Complex multiplication

The elliptic curves in class 8400bt do not have complex multiplication.

## Modular form8400.2.a.bt

sage: E.q_eigenform(10)

$$q - q^{3} + q^{7} + q^{9} + 2 q^{11} - 2 q^{13} - 8 q^{17} + 2 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.