# Properties

 Label 100800cx Number of curves $8$ Conductor $100800$ CM no Rank $0$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("100800.ec1")

sage: E.isogeny_class()

## Elliptic curves in class 100800cx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100800.ec7 100800cx1 [0, 0, 0, -590700, 61486000] [2] 1769472 $$\Gamma_0(N)$$-optimal
100800.ec5 100800cx2 [0, 0, 0, -5198700, -4518866000] [2, 2] 3538944
100800.ec4 100800cx3 [0, 0, 0, -38606700, 92329774000] [2] 5308416
100800.ec6 100800cx4 [0, 0, 0, -1166700, -11349074000] [2] 7077888
100800.ec2 100800cx5 [0, 0, 0, -82958700, -290831186000] [2] 7077888
100800.ec3 100800cx6 [0, 0, 0, -38894700, 90882286000] [2, 2] 10616832
100800.ec8 100800cx7 [0, 0, 0, 10497300, 305935054000] [2] 21233664
100800.ec1 100800cx8 [0, 0, 0, -92894700, -216809714000] [2] 21233664

## Rank

sage: E.rank()

The elliptic curves in class 100800cx have rank $$0$$.

## Modular form 100800.2.a.ec

sage: E.q_eigenform(10)

$$q - q^{7} + 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.