# Properties

 Label 235200ms Number of curves $6$ Conductor $235200$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("235200.ms1")

sage: E.isogeny_class()

## Elliptic curves in class 235200ms

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
235200.ms6 235200ms1 [0, -1, 0, 782367, -373012863] [2] 7077888 $$\Gamma_0(N)$$-optimal
235200.ms5 235200ms2 [0, -1, 0, -5489633, -3853972863] [2, 2] 14155776
235200.ms4 235200ms3 [0, -1, 0, -29009633, 56851147137] [2] 28311552
235200.ms2 235200ms4 [0, -1, 0, -82321633, -287440884863] [2, 2] 28311552
235200.ms3 235200ms5 [0, -1, 0, -76833633, -327420964863] [2] 56623104
235200.ms1 235200ms6 [0, -1, 0, -1317121633, -18398252484863] [2] 56623104

## Rank

sage: E.rank()

The elliptic curves in class 235200ms have rank $$1$$.

## Modular form 235200.2.a.ms

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} + 4q^{11} + 2q^{13} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.