# Properties

 Label 12705n Number of curves 4 Conductor 12705 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12705.g1")

sage: E.isogeny_class()

## Elliptic curves in class 12705n

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12705.g3 12705n1 [1, 0, 0, -305, -1968]  5760 $$\Gamma_0(N)$$-optimal
12705.g2 12705n2 [1, 0, 0, -910, 8075] [2, 2] 11520
12705.g1 12705n3 [1, 0, 0, -13615, 610292]  23040
12705.g4 12705n4 [1, 0, 0, 2115, 51030]  23040

## Rank

sage: E.rank()

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

## Modular form 12705.2.a.g

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 