# Properties

 Label 344850.gz Number of curves $4$ Conductor $344850$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("gz1")

sage: E.isogeny_class()

## Elliptic curves in class 344850.gz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
344850.gz1 344850gz3 $$[1, 0, 0, -1878588, 989452542]$$ $$26487576322129/44531250$$ $$1232653527832031250$$ $$[2]$$ $$7864320$$ $$2.3666$$
344850.gz2 344850gz2 $$[1, 0, 0, -154338, 4905792]$$ $$14688124849/8122500$$ $$224836003476562500$$ $$[2, 2]$$ $$3932160$$ $$2.0200$$
344850.gz3 344850gz1 $$[1, 0, 0, -93838, -11005708]$$ $$3301293169/22800$$ $$631118606250000$$ $$[2]$$ $$1966080$$ $$1.6734$$ $$\Gamma_0(N)$$-optimal
344850.gz4 344850gz4 $$[1, 0, 0, 601912, 38937042]$$ $$871257511151/527800050$$ $$-14609843505907031250$$ $$[2]$$ $$7864320$$ $$2.3666$$

## Rank

sage: E.rank()

The elliptic curves in class 344850.gz have rank $$1$$.

## Complex multiplication

The elliptic curves in class 344850.gz do not have complex multiplication.

## Modular form 344850.2.a.gz

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + q^{12} + 2q^{13} + q^{16} + 2q^{17} + q^{18} + q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.