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Properties

Label 10.2.9130086859014144.15
Degree $10$
Signature $[2, 4]$
Discriminant $9.130\times 10^{15}$
Root discriminant \(39.45\)
Ramified primes $2,3$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $M_{10}$ (as 10T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184)
 
gp: K = bnfinit(y^10 - 4*y^9 + 12*y^8 - 96*y^6 + 144*y^5 + 120*y^4 - 528*y^3 + 798*y^2 - 856*y + 184, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184)
 

\( x^{10} - 4x^{9} + 12x^{8} - 96x^{6} + 144x^{5} + 120x^{4} - 528x^{3} + 798x^{2} - 856x + 184 \) 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:   \(9130086859014144\) \(\medspace = 2^{34}\cdot 3^{12}\) 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:  \(39.45\)
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:  $2^{31/8}3^{25/18}\approx 67.47760233783548$
Ramified primes:   \(2\), \(3\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{138457690080}a^{9}+\frac{394494297}{23076281680}a^{8}-\frac{1099423789}{5769070420}a^{7}-\frac{835628687}{2884535210}a^{6}-\frac{266441182}{1442267605}a^{5}+\frac{1364946959}{2884535210}a^{4}+\frac{2120256413}{5769070420}a^{3}+\frac{395591374}{1442267605}a^{2}-\frac{593848123}{23076281680}a+\frac{17038094069}{34614422520}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{4}$, which has order $4$ (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:   $\frac{456782383}{138457690080}a^{9}-\frac{242932569}{23076281680}a^{8}+\frac{260719053}{5769070420}a^{7}-\frac{62280071}{2884535210}a^{6}-\frac{134512501}{1442267605}a^{5}+\frac{259626577}{2884535210}a^{4}+\frac{1661030039}{5769070420}a^{3}-\frac{771768563}{1442267605}a^{2}+\frac{22756214891}{23076281680}a-\frac{10473210373}{34614422520}$, $\frac{283095707}{46152563360}a^{9}-\frac{425422823}{23076281680}a^{8}+\frac{198784511}{5769070420}a^{7}+\frac{344211953}{2884535210}a^{6}-\frac{1003691532}{1442267605}a^{5}+\frac{289224049}{2884535210}a^{4}+\frac{18610664053}{5769070420}a^{3}-\frac{5103562421}{1442267605}a^{2}-\frac{34697643323}{23076281680}a+\frac{12910521143}{11538140840}$, $\frac{1253550793}{46152563360}a^{9}-\frac{2429690237}{23076281680}a^{8}+\frac{1715652209}{5769070420}a^{7}+\frac{176423257}{2884535210}a^{6}-\frac{3918661273}{1442267605}a^{5}+\frac{9638797731}{2884535210}a^{4}+\frac{26277859207}{5769070420}a^{3}-\frac{19256951424}{1442267605}a^{2}+\frac{440139259063}{23076281680}a-\frac{212068794723}{11538140840}$, $\frac{592571797}{34614422520}a^{9}-\frac{381110291}{5769070420}a^{8}+\frac{288411647}{1442267605}a^{7}+\frac{35213762}{1442267605}a^{6}-\frac{2381622156}{1442267605}a^{5}+\frac{3467607221}{1442267605}a^{4}+\frac{3379112316}{1442267605}a^{3}-\frac{13805301058}{1442267605}a^{2}+\frac{75507749389}{5769070420}a-\frac{94725583897}{8653605630}$, $\frac{8699221}{4615256336}a^{9}-\frac{9736529}{2307628168}a^{8}+\frac{4286055}{576907042}a^{7}+\frac{8507334}{288453521}a^{6}-\frac{28542233}{288453521}a^{5}-\frac{17953958}{288453521}a^{4}+\frac{129015785}{576907042}a^{3}-\frac{78532271}{288453521}a^{2}+\frac{1050076795}{2307628168}a-\frac{374318699}{1153814084}$ 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:  \( 22631.9020885 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 22631.9020885 \cdot 4}{2\cdot\sqrt{9130086859014144}}\cr\approx \mathstrut & 2.95320137122 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184)
 
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 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184, 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 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184);
 
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 - 4*x^9 + 12*x^8 - 96*x^6 + 144*x^5 + 120*x^4 - 528*x^3 + 798*x^2 - 856*x + 184);
 
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

$A_6.C_2$ (as 10T31):

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 720
The 8 conjugacy class representatives for $M_{10}$
Character table for $M_{10}$

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 12 sibling: data not computed
Degree 20 siblings: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.3.1$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$[3]$
2.8.31.2$x^{8} + 16 x^{7} + 8 x^{6} + 16 x^{5} + 4 x^{4} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
\(3\) 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}$