# Properties

 Label 31200d Number of curves $4$ Conductor $31200$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 31200d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31200.k3 31200d1 $$[0, -1, 0, -1658, -23688]$$ $$504358336/38025$$ $$38025000000$$ $$[2, 2]$$ $$18432$$ $$0.77535$$ $$\Gamma_0(N)$$-optimal
31200.k4 31200d2 $$[0, -1, 0, 1592, -108188]$$ $$55742968/658125$$ $$-5265000000000$$ $$[2]$$ $$36864$$ $$1.1219$$
31200.k2 31200d3 $$[0, -1, 0, -5408, 126312]$$ $$2186875592/428415$$ $$3427320000000$$ $$[2]$$ $$36864$$ $$1.1219$$
31200.k1 31200d4 $$[0, -1, 0, -26033, -1608063]$$ $$30488290624/195$$ $$12480000000$$ $$[2]$$ $$36864$$ $$1.1219$$

## Rank

sage: E.rank()

The elliptic curves in class 31200d have rank $$0$$.

## Complex multiplication

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

## Modular form 31200.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} + q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.