# Properties

 Label 141120.md Number of curves $8$ Conductor $141120$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("141120.md1")

sage: E.isogeny_class()

## Elliptic curves in class 141120.md

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
141120.md1 141120bd8 [0, 0, 0, -182073612, -594925855216] [2] 42467328
141120.md2 141120bd5 [0, 0, 0, -162599052, -798040774384] [2] 14155776
141120.md3 141120bd6 [0, 0, 0, -76233612, 249380992784] [2, 2] 21233664
141120.md4 141120bd3 [0, 0, 0, -75669132, 253352899856] [2] 10616832
141120.md5 141120bd2 [0, 0, 0, -10189452, -12399768304] [2, 2] 7077888
141120.md6 141120bd4 [0, 0, 0, -2286732, -31141859056] [2] 14155776
141120.md7 141120bd1 [0, 0, 0, -1157772, 168717584] [2] 3538944 $$\Gamma_0(N)$$-optimal
141120.md8 141120bd7 [0, 0, 0, 20574708, 839485788176] [2] 42467328

## Rank

sage: E.rank()

The elliptic curves in class 141120.md have rank $$1$$.

## Modular form 141120.2.a.md

sage: E.q_eigenform(10)

$$q + q^{5} + 2q^{13} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.