# Properties

 Label 6450.bc Number of curves $2$ Conductor $6450$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bc1")

sage: E.isogeny_class()

## Elliptic curves in class 6450.bc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6450.bc1 6450bb2 $$[1, 1, 1, -5938, -172969]$$ $$1481933914201/53916840$$ $$842450625000$$ $$$$ $$13824$$ $$1.0580$$
6450.bc2 6450bb1 $$[1, 1, 1, -938, 7031]$$ $$5841725401/1857600$$ $$29025000000$$ $$$$ $$6912$$ $$0.71145$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 6450.bc have rank $$1$$.

## Complex multiplication

The elliptic curves in class 6450.bc do not have complex multiplication.

## Modular form6450.2.a.bc

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + 2q^{7} + q^{8} + q^{9} - 2q^{11} - q^{12} + 2q^{13} + 2q^{14} + q^{16} + 4q^{17} + q^{18} - 6q^{19} + O(q^{20})$$ ## 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. 