# Properties

 Label 61347z Number of curves $6$ Conductor $61347$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("61347.i1")

sage: E.isogeny_class()

## Elliptic curves in class 61347z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
61347.i5 61347z1 [1, 0, 0, -491202, -190117773] [4] 1290240 $$\Gamma_0(N)$$-optimal
61347.i4 61347z2 [1, 0, 0, -8773047, -10000791360] [2, 2] 2580480
61347.i3 61347z3 [1, 0, 0, -9693252, -7774815465] [2, 2] 5160960
61347.i1 61347z4 [1, 0, 0, -140362362, -640076749443] [2] 5160960
61347.i6 61347z5 [1, 0, 0, 27421683, -52973383308] [2] 10321920
61347.i2 61347z6 [1, 0, 0, -61531467, 179889890478] [2] 10321920

## Rank

sage: E.rank()

The elliptic curves in class 61347z have rank $$1$$.

## Modular form 61347.2.a.i

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + 3q^{8} + q^{9} - 2q^{10} - q^{12} + 2q^{15} - q^{16} + 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.