↑

Properties

Label 10.2.34828517376000000.40
Degree $10$
Signature $[2, 4]$
Discriminant $3.483\times 10^{16}$
Root discriminant \(45.10\)
Ramified primes $2,3,5$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $\PSL(2,9)$ (as 10T26)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440)
 
gp: K = bnfinit(y^10 - 2*y^9 - 3*y^8 + 78*y^6 - 156*y^5 + 30*y^4 - 384*y^3 + 576*y^2 + 208*y - 440, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440)
 

\( x^{10} - 2x^{9} - 3x^{8} + 78x^{6} - 156x^{5} + 30x^{4} - 384x^{3} + 576x^{2} + 208x - 440 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(34828517376000000\) \(\medspace = 2^{22}\cdot 3^{12}\cdot 5^{6}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(45.10\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{11/4}3^{25/18}5^{2/3}\approx 90.46506515166082$
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{8}a^{8}+\frac{1}{8}a^{6}+\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{3018132336}a^{9}+\frac{25887803}{503022056}a^{8}+\frac{37211061}{1006044112}a^{7}-\frac{16291737}{251511028}a^{6}+\frac{4821488}{62877757}a^{5}+\frac{100439749}{251511028}a^{4}+\frac{155607321}{503022056}a^{3}-\frac{33383607}{125755514}a^{2}-\frac{15627609}{251511028}a-\frac{41689028}{188633271}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1829479}{1509066168}a^{9}-\frac{1776647}{503022056}a^{8}-\frac{12548083}{503022056}a^{7}+\frac{9021253}{503022056}a^{6}+\frac{9808014}{62877757}a^{5}-\frac{36429861}{125755514}a^{4}-\frac{353623423}{251511028}a^{3}+\frac{328835817}{251511028}a^{2}+\frac{7866675}{125755514}a-\frac{27204077}{377266542}$, $\frac{4258355}{503022056}a^{9}+\frac{14934291}{503022056}a^{8}-\frac{102964537}{503022056}a^{7}+\frac{6713799}{503022056}a^{6}+\frac{156279949}{125755514}a^{5}+\frac{193685663}{125755514}a^{4}-\frac{3624298525}{251511028}a^{3}+\frac{3231880959}{251511028}a^{2}+\frac{828791769}{125755514}a-\frac{1113201493}{125755514}$, $\frac{294601729}{1509066168}a^{9}-\frac{139980647}{251511028}a^{8}-\frac{8033799}{503022056}a^{7}-\frac{7774505}{125755514}a^{6}+\frac{911818591}{62877757}a^{5}-\frac{5338897985}{125755514}a^{4}+\frac{12695630609}{251511028}a^{3}-\frac{7966491759}{62877757}a^{2}+\frac{23318770389}{125755514}a-\frac{16856899681}{188633271}$, $\frac{38069707}{1006044112}a^{9}-\frac{1735682}{62877757}a^{8}-\frac{137497395}{1006044112}a^{7}-\frac{106202649}{503022056}a^{6}+\frac{164710904}{62877757}a^{5}-\frac{596466983}{251511028}a^{4}-\frac{425191887}{503022056}a^{3}-\frac{4760013041}{251511028}a^{2}-\frac{927986413}{251511028}a+\frac{1235918429}{125755514}$, $\frac{9914746463}{3018132336}a^{9}+\frac{205350229}{125755514}a^{8}-\frac{5798902965}{1006044112}a^{7}-\frac{7245465511}{503022056}a^{6}+\frac{13875067093}{62877757}a^{5}+\frac{9379554327}{251511028}a^{4}+\frac{97597275535}{503022056}a^{3}-\frac{195026926551}{251511028}a^{2}-\frac{10092363195}{251511028}a+\frac{218970082237}{377266542}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 344083.411284 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 344083.411284 \cdot 2}{2\cdot\sqrt{34828517376000000}}\cr\approx \mathstrut & 11.4941194558 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 - 3*x^8 + 78*x^6 - 156*x^5 + 30*x^4 - 384*x^3 + 576*x^2 + 208*x - 440);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$A_6$ (as 10T26):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $\PSL(2,9)$
Character table for $\PSL(2,9)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 6 siblings: 6.2.7464960000.5, some data not computed
Degree 15 siblings: data not computed
Degree 20 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.2.298598400.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.9.12.21$x^{9} + 6 x^{4} + 3$$9$$1$$12$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.3.2.1$x^{3} + 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$