# Properties

 Label 324870el Number of curves 8 Conductor 324870 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("324870.el1")

sage: E.isogeny_class()

## Elliptic curves in class 324870el

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
324870.el7 324870el1 [1, 1, 1, -102711155, -385844310223] [2] 79626240 $$\Gamma_0(N)$$-optimal
324870.el6 324870el2 [1, 1, 1, -272306035, 1216420278065] [2, 2] 159252480
324870.el5 324870el3 [1, 1, 1, -1263846515, 17180605977905] [2] 238878720
324870.el4 324870el4 [1, 1, 1, -3987290035, 96895090201265] [2] 318504960
324870.el8 324870el5 [1, 1, 1, 729159885, 8086075902897] [2] 318504960
324870.el2 324870el6 [1, 1, 1, -20184937795, 1103788604442257] [2, 2] 477757440
324870.el1 324870el7 [1, 1, 1, -322959000295, 70642876415817257] [2] 955514880
324870.el3 324870el8 [1, 1, 1, -20148335775, 1107991116611385] [2] 955514880

## Rank

sage: E.rank()

The elliptic curves in class 324870el have rank $$0$$.

## Modular form 324870.2.a.el

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.