# Properties

 Label 31200.s Number of curves $2$ Conductor $31200$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

## Elliptic curves in class 31200.s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31200.s1 31200bo1 $$[0, -1, 0, -264758, 52428012]$$ $$2052450196928704/4317958125$$ $$4317958125000000$$ $$[2]$$ $$221184$$ $$1.8852$$ $$\Gamma_0(N)$$-optimal
31200.s2 31200bo2 $$[0, -1, 0, -173633, 88969137]$$ $$-9045718037056/48125390625$$ $$-3080025000000000000$$ $$[2]$$ $$442368$$ $$2.2318$$

## Rank

sage: E.rank()

The elliptic curves in class 31200.s have rank $$1$$.

## Complex multiplication

The elliptic curves in class 31200.s do not have complex multiplication.

## Modular form 31200.2.a.s

sage: E.q_eigenform(10)

$$q - q^{3} + 2 q^{7} + q^{9} - 2 q^{11} + q^{13} + 2 q^{17} - 2 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.