# Properties

 Label 27600ch Number of curves $6$ Conductor $27600$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27600.cn1")

sage: E.isogeny_class()

## Elliptic curves in class 27600ch

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27600.cn5 27600ch1 [0, 1, 0, -168008, -29136012]  221184 $$\Gamma_0(N)$$-optimal
27600.cn4 27600ch2 [0, 1, 0, -2760008, -1765776012] [2, 2] 442368
27600.cn3 27600ch3 [0, 1, 0, -2832008, -1668864012] [2, 2] 884736
27600.cn1 27600ch4 [0, 1, 0, -44160008, -112966176012]  884736
27600.cn6 27600ch5 [0, 1, 0, 3515992, -8080344012]  1769472
27600.cn2 27600ch6 [0, 1, 0, -10332008, 10946135988]  1769472

## Rank

sage: E.rank()

The elliptic curves in class 27600ch have rank $$1$$.

## Modular form 27600.2.a.cn

sage: E.q_eigenform(10)

$$q + q^{3} + q^{9} - 4q^{11} + 2q^{13} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 