# Properties

 Label 331200.hr Number of curves $6$ Conductor $331200$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("331200.hr1")

sage: E.isogeny_class()

## Elliptic curves in class 331200.hr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
331200.hr1 331200hr3 [0, 0, 0, -1589760300, 24397514498000] [2] 56623104
331200.hr2 331200hr6 [0, 0, 0, -371952300, -2365109278000] [2] 113246208
331200.hr3 331200hr4 [0, 0, 0, -101952300, 360270722000] [2, 2] 56623104
331200.hr4 331200hr2 [0, 0, 0, -99360300, 381208898000] [2, 2] 28311552
331200.hr5 331200hr1 [0, 0, 0, -6048300, 6281282000] [2] 14155776 $$\Gamma_0(N)$$-optimal
331200.hr6 331200hr5 [0, 0, 0, 126575700, 1745607458000] [2] 113246208

## Rank

sage: E.rank()

The elliptic curves in class 331200.hr have rank $$2$$.

## Modular form 331200.2.a.hr

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.