# Properties

 Label 4410m Number of curves $4$ Conductor $4410$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("m1")

sage: E.isogeny_class()

## Elliptic curves in class 4410m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4410.b4 4410m1 $$[1, -1, 0, 1020, -20224]$$ $$1367631/2800$$ $$-240145138800$$ $$$$ $$6144$$ $$0.86810$$ $$\Gamma_0(N)$$-optimal
4410.b3 4410m2 $$[1, -1, 0, -7800, -212500]$$ $$611960049/122500$$ $$10506349822500$$ $$[2, 2]$$ $$12288$$ $$1.2147$$
4410.b1 4410m3 $$[1, -1, 0, -118050, -15581350]$$ $$2121328796049/120050$$ $$10296222826050$$ $$$$ $$24576$$ $$1.5612$$
4410.b2 4410m4 $$[1, -1, 0, -38670, 2744846]$$ $$74565301329/5468750$$ $$469033474218750$$ $$$$ $$24576$$ $$1.5612$$

## Rank

sage: E.rank()

The elliptic curves in class 4410m have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4410m do not have complex multiplication.

## Modular form4410.2.a.m

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 4q^{11} + 6q^{13} + q^{16} + 2q^{17} + O(q^{20})$$ ## 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 Cremona 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 Cremona labels. 