# Properties

 Label 304200.fw Number of curves $4$ Conductor $304200$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("fw1")

sage: E.isogeny_class()

## Elliptic curves in class 304200.fw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
304200.fw1 304200fw4 $$[0, 0, 0, -316380675, -2166022849250]$$ $$31103978031362/195$$ $$21956961068640000000$$ $$[2]$$ $$49545216$$ $$3.3169$$
304200.fw2 304200fw3 $$[0, 0, 0, -27390675, -5427139250]$$ $$20183398562/11567205$$ $$1302464973630656160000000$$ $$[2]$$ $$49545216$$ $$3.3169$$
304200.fw3 304200fw2 $$[0, 0, 0, -19785675, -33801394250]$$ $$15214885924/38025$$ $$2140803704192400000000$$ $$[2, 2]$$ $$24772608$$ $$2.9703$$
304200.fw4 304200fw1 $$[0, 0, 0, -773175, -928781750]$$ $$-3631696/24375$$ $$-343077516697500000000$$ $$[2]$$ $$12386304$$ $$2.6237$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 304200.fw have rank $$0$$.

## Complex multiplication

The elliptic curves in class 304200.fw do not have complex multiplication.

## Modular form 304200.2.a.fw

sage: E.q_eigenform(10)

$$q + 4q^{7} + 4q^{11} + 6q^{17} + O(q^{20})$$

## 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.