# Properties

 Label 4400.e Number of curves $4$ Conductor $4400$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 4400.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4400.e1 4400t4 $$[0, 1, 0, -177508, 28726488]$$ $$154639330142416/33275$$ $$133100000000$$ $$$$ $$20736$$ $$1.5192$$
4400.e2 4400t3 $$[0, 1, 0, -11133, 442738]$$ $$610462990336/8857805$$ $$2214451250000$$ $$$$ $$10368$$ $$1.1726$$
4400.e3 4400t2 $$[0, 1, 0, -2508, 26488]$$ $$436334416/171875$$ $$687500000000$$ $$$$ $$6912$$ $$0.96986$$
4400.e4 4400t1 $$[0, 1, 0, -1133, -14762]$$ $$643956736/15125$$ $$3781250000$$ $$$$ $$3456$$ $$0.62329$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 4400.e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4400.e do not have complex multiplication.

## Modular form4400.2.a.e

sage: E.q_eigenform(10)

$$q - 2 q^{3} - 4 q^{7} + q^{9} + q^{11} + 4 q^{13} + 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 