# Properties

 Label 133200cs Number of curves $6$ Conductor $133200$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("133200.dz1")

sage: E.isogeny_class()

## Elliptic curves in class 133200cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
133200.dz5 133200cs1 [0, 0, 0, -3042075, 7030210250] [2] 7962624 $$\Gamma_0(N)$$-optimal
133200.dz4 133200cs2 [0, 0, 0, -76770075, 258368962250] [2, 2] 15925248
133200.dz1 133200cs3 [0, 0, 0, -1227618075, 16555527490250] [2] 31850496
133200.dz3 133200cs4 [0, 0, 0, -105570075, 46890562250] [2, 2] 31850496
133200.dz6 133200cs5 [0, 0, 0, 419309925, 373890802250] [2] 63700992
133200.dz2 133200cs6 [0, 0, 0, -1091250075, -13814727277750] [2] 63700992

## Rank

sage: E.rank()

The elliptic curves in class 133200cs have rank $$1$$.

## Modular form 133200.2.a.dz

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.