# Properties

 Label 47040ec Number of curves $8$ Conductor $47040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.z1")

sage: E.isogeny_class()

## Elliptic curves in class 47040ec

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.z7 47040ec1 [0, -1, 0, -128641, -6205919] [2] 442368 $$\Gamma_0(N)$$-optimal
47040.z5 47040ec2 [0, -1, 0, -1132161, 459628065] [2, 2] 884736
47040.z4 47040ec3 [0, -1, 0, -8407681, -9380638175] [2] 1327104
47040.z6 47040ec4 [0, -1, 0, -254081, 1153486881] [2] 1769472
47040.z2 47040ec5 [0, -1, 0, -18066561, 29563087905] [2] 1769472
47040.z3 47040ec6 [0, -1, 0, -8470401, -9233509599] [2, 2] 2654208
47040.z8 47040ec7 [0, -1, 0, 2286079, -31092828255] [2] 5308416
47040.z1 47040ec8 [0, -1, 0, -20230401, 22041034401] [2] 5308416

## Rank

sage: E.rank()

The elliptic curves in class 47040ec have rank $$1$$.

## Modular form 47040.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} + 2q^{13} + q^{15} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.