# Properties

 Label 100800fn Number of curves $4$ Conductor $100800$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("100800.op1")

sage: E.isogeny_class()

## Elliptic curves in class 100800fn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
100800.op4 100800fn1 [0, 0, 0, 33300, 3726000]  589824 $$\Gamma_0(N)$$-optimal
100800.op3 100800fn2 [0, 0, 0, -254700, 40014000] [2, 2] 1179648
100800.op2 100800fn3 [0, 0, 0, -1262700, -510354000]  2359296
100800.op1 100800fn4 [0, 0, 0, -3854700, 2912814000]  2359296

## Rank

sage: E.rank()

The elliptic curves in class 100800fn have rank $$1$$.

## Modular form 100800.2.a.op

sage: E.q_eigenform(10)

$$q + q^{7} + 4q^{11} - 6q^{13} + 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 