# Properties

 Label 106470ea Number of curves $8$ Conductor $106470$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("106470.dr1")

sage: E.isogeny_class()

## Elliptic curves in class 106470ea

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106470.dr7 106470ea1 [1, -1, 1, -62393, 2126297] [2] 884736 $$\Gamma_0(N)$$-optimal
106470.dr5 106470ea2 [1, -1, 1, -549113, -154986919] [2, 2] 1769472
106470.dr4 106470ea3 [1, -1, 1, -4077833, 3170527481] [2] 2654208
106470.dr6 106470ea4 [1, -1, 1, -123233, -389561623] [2] 3538944
106470.dr2 106470ea5 [1, -1, 1, -8762513, -9981498679] [2] 3538944
106470.dr3 106470ea6 [1, -1, 1, -4108253, 3120845537] [2, 2] 5308416
106470.dr8 106470ea7 [1, -1, 1, 1108777, 10501899581] [2] 10616832
106470.dr1 106470ea8 [1, -1, 1, -9812003, -7440217963] [2] 10616832

## Rank

sage: E.rank()

The elliptic curves in class 106470ea have rank $$1$$.

## Modular form 106470.2.a.dr

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} - q^{14} + q^{16} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.