# Properties

 Label 8400cf Number of curves 8 Conductor 8400 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("8400.cm1")

sage: E.isogeny_class()

## Elliptic curves in class 8400cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8400.cm7 8400cf1 [0, 1, 0, -199008, -34224012] [2] 55296 $$\Gamma_0(N)$$-optimal
8400.cm6 8400cf2 [0, 1, 0, -231008, -22512012] [2, 2] 110592
8400.cm5 8400cf3 [0, 1, 0, -589008, 132035988] [2] 165888
8400.cm4 8400cf4 [0, 1, 0, -1743008, 869567988] [2] 221184
8400.cm8 8400cf5 [0, 1, 0, 768992, -164512012] [4] 221184
8400.cm2 8400cf6 [0, 1, 0, -8781008, 10011587988] [2, 2] 331776
8400.cm1 8400cf7 [0, 1, 0, -140493008, 640912067988] [2] 663552
8400.cm3 8400cf8 [0, 1, 0, -8141008, 11533507988] [4] 663552

## Rank

sage: E.rank()

The elliptic curves in class 8400cf have rank $$0$$.

## Modular form8400.2.a.cm

sage: E.q_eigenform(10)

$$q + q^{3} + q^{7} + q^{9} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.