# Properties

 Label 88200.hm Number of curves $4$ Conductor $88200$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hm1")

sage: E.isogeny_class()

## Elliptic curves in class 88200.hm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
88200.hm1 88200gt4 [0, 0, 0, -3296475, 2303673750]  1572864
88200.hm2 88200gt3 [0, 0, 0, -650475, -159752250]  1572864
88200.hm3 88200gt2 [0, 0, 0, -209475, 34728750] [2, 2] 786432
88200.hm4 88200gt1 [0, 0, 0, 11025, 2315250]  393216 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 88200.hm have rank $$0$$.

## Complex multiplication

The elliptic curves in class 88200.hm do not have complex multiplication.

## Modular form 88200.2.a.hm

sage: E.q_eigenform(10)

$$q + 4q^{11} + 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 