# Properties

 Label 39270.cp Number of curves 8 Conductor 39270 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("39270.cp1")
sage: E.isogeny_class()

## Elliptic curves in class 39270.cp

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
39270.cp1 39270cn8 [1, 0, 0, -1333355333356, -592607370447828664] 2 222953472
39270.cp2 39270cn6 [1, 0, 0, -83334708356, -9259495366703664] 4 111476736
39270.cp3 39270cn7 [1, 0, 0, -83314083356, -9264307785578664] 2 222953472
39270.cp4 39270cn5 [1, 0, 0, -16461218116, -812901436975600] 6 74317824
39270.cp5 39270cn3 [1, 0, 0, -5209708356, -144604741703664] 2 55738368
39270.cp6 39270cn2 [1, 0, 0, -1054050116, -12046088636400] 12 37158912
39270.cp7 39270cn1 [1, 0, 0, -234850116, 1175307843600] 6 18579456 $$\Gamma_0(N)$$-optimal
39270.cp8 39270cn4 [1, 0, 0, 1245917884, -57356838217200] 6 74317824

## Rank

sage: E.rank()

The elliptic curves in class 39270.cp have rank $$0$$.

## Modular form None

sage: E.q_eigenform(10)
$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + 2q^{13} + q^{14} - q^{15} + q^{16} + q^{17} + q^{18} + 8q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.