# Properties

 Label 405600.g Number of curves $4$ Conductor $405600$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 405600.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.g1 405600g2 $$[0, -1, 0, -961273408, 11468232029812]$$ $$2543984126301795848/909361981125$$ $$35114532758015841000000000$$ $$[2]$$ $$198180864$$ $$3.8710$$ $$\Gamma_0(N)$$-optimal*
405600.g2 405600g4 $$[0, -1, 0, -496523408, -4170763907688]$$ $$350584567631475848/8259273550125$$ $$318927487241642409000000000$$ $$[2]$$ $$198180864$$ $$3.8710$$
405600.g3 405600g1 $$[0, -1, 0, -68742158, 124159842312]$$ $$7442744143086784/2927948765625$$ $$14132649453457640625000000$$ $$[2, 2]$$ $$99090432$$ $$3.5244$$ $$\Gamma_0(N)$$-optimal*
405600.g4 405600g3 $$[0, -1, 0, 220437967, 895981595937]$$ $$3834800837445824/3342041015625$$ $$-1032409193765625000000000000$$ $$[2]$$ $$198180864$$ $$3.8710$$
*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.g1.

## Rank

sage: E.rank()

The elliptic curves in class 405600.g have rank $$0$$.

## Complex multiplication

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

## Modular form 405600.2.a.g

sage: E.q_eigenform(10)

$$q - q^{3} - 4q^{7} + q^{9} - 2q^{17} + 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.