# Properties

 Label 2200.e Number of curves $4$ Conductor $2200$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 2200.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2200.e1 2200f3 $$[0, 0, 0, -188675, -1623250]$$ $$46424454082884/26794860125$$ $$428717762000000000$$ $$[2]$$ $$27648$$ $$2.0728$$
2200.e2 2200f2 $$[0, 0, 0, -126175, 17189250]$$ $$55537159171536/228765625$$ $$915062500000000$$ $$[2, 2]$$ $$13824$$ $$1.7263$$
2200.e3 2200f1 $$[0, 0, 0, -126050, 17225125]$$ $$885956203616256/15125$$ $$3781250000$$ $$[4]$$ $$6912$$ $$1.3797$$ $$\Gamma_0(N)$$-optimal
2200.e4 2200f4 $$[0, 0, 0, -65675, 33705750]$$ $$-1957960715364/29541015625$$ $$-472656250000000000$$ $$[2]$$ $$27648$$ $$2.0728$$

## Rank

sage: E.rank()

The elliptic curves in class 2200.e have rank $$0$$.

## Complex multiplication

The elliptic curves in class 2200.e do not have complex multiplication.

## Modular form2200.2.a.e

sage: E.q_eigenform(10)

$$q - 4 q^{7} - 3 q^{9} - q^{11} - 6 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})$$

## 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.