# Properties

 Label 405600.hd Number of curves $2$ Conductor $405600$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hd1")

sage: E.isogeny_class()

## Elliptic curves in class 405600.hd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.hd1 405600hd2 $$[0, 1, 0, -1049208, -135119412]$$ $$26463592/13689$$ $$66074188401000000000$$ $$[2]$$ $$12042240$$ $$2.4951$$ $$\Gamma_0(N)$$-optimal*
405600.hd2 405600hd1 $$[0, 1, 0, -837958, -295246912]$$ $$107850176/117$$ $$70592081625000000$$ $$[2]$$ $$6021120$$ $$2.1486$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 405600.hd1.

## Rank

sage: E.rank()

The elliptic curves in class 405600.hd have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405600.hd do not have complex multiplication.

## Modular form 405600.2.a.hd

sage: E.q_eigenform(10)

$$q + q^{3} + 4q^{7} + q^{9} + 2q^{11} - 2q^{19} + O(q^{20})$$

## 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.