# Properties

 Label 109200fz Number of curves 8 Conductor 109200 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("109200.ge1")

sage: E.isogeny_class()

## Elliptic curves in class 109200fz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
109200.ge7 109200fz1 [0, 1, 0, -10290008, -12640236012] [2] 5308416 $$\Gamma_0(N)$$-optimal
109200.ge6 109200fz2 [0, 1, 0, -16562008, 4582675988] [2, 2] 10616832
109200.ge5 109200fz3 [0, 1, 0, -63570008, 186566483988] [2] 15925248
109200.ge8 109200fz4 [0, 1, 0, 65085992, 36425395988] [2] 21233664
109200.ge4 109200fz5 [0, 1, 0, -198562008, 1075106675988] [2] 21233664
109200.ge2 109200fz6 [0, 1, 0, -1004762008, 12258295075988] [2, 2] 31850496
109200.ge3 109200fz7 [0, 1, 0, -992414008, 12574280395988] [2] 63700992
109200.ge1 109200fz8 [0, 1, 0, -16076182008, 784547998715988] [2] 63700992

## Rank

sage: E.rank()

The elliptic curves in class 109200fz have rank $$0$$.

## Modular form 109200.2.a.ge

sage: E.q_eigenform(10)

$$q + q^{3} + q^{7} + q^{9} - q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.