# Properties

 Label 12705.l Number of curves $6$ Conductor $12705$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12705.l1")

sage: E.isogeny_class()

## Elliptic curves in class 12705.l

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12705.l1 12705g5 [1, 1, 0, -1603252, 780654241]  245760
12705.l2 12705g3 [1, 1, 0, -105877, 10704016] [2, 2] 122880
12705.l3 12705g2 [1, 1, 0, -32672, -2136141] [2, 2] 61440
12705.l4 12705g1 [1, 1, 0, -32067, -2223624]  30720 $$\Gamma_0(N)$$-optimal
12705.l5 12705g4 [1, 1, 0, 30853, -9365286]  122880
12705.l6 12705g6 [1, 1, 0, 220218, 63987939]  245760

## Rank

sage: E.rank()

The elliptic curves in class 12705.l have rank $$0$$.

## Modular form 12705.2.a.l

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + q^{7} - 3q^{8} + q^{9} + q^{10} + q^{12} + 2q^{13} + q^{14} - q^{15} - q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 