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Properties

Label 10.4.3690562500000000.10
Degree $10$
Signature $[4, 3]$
Discriminant $-3.691\times 10^{15}$
Root discriminant \(36.03\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_5^2 \wr C_2$ (as 10T43)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359)
 
gp: K = bnfinit(y^10 + 5*y^8 + 10*y^6 - 228*y^5 + 90*y^4 + 600*y^3 - 1755*y^2 + 2700*y - 1359, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359)
 

\( x^{10} + 5x^{8} + 10x^{6} - 228x^{5} + 90x^{4} + 600x^{3} - 1755x^{2} + 2700x - 1359 \) 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:  $[4, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-3690562500000000\) \(\medspace = -\,2^{8}\cdot 3^{10}\cdot 5^{12}\) 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:  \(36.03\)
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:  $2^{5/3}3^{11/6}5^{271/200}\approx 210.64119776948417$
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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{24}a^{7}-\frac{1}{8}a^{6}-\frac{1}{24}a^{5}+\frac{1}{8}a^{4}-\frac{5}{24}a^{3}+\frac{1}{8}a^{2}+\frac{3}{8}a+\frac{3}{8}$, $\frac{1}{48}a^{8}+\frac{1}{24}a^{6}+\frac{1}{12}a^{4}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a+\frac{1}{16}$, $\frac{1}{546917664}a^{9}-\frac{1286995}{182305888}a^{8}+\frac{3565999}{273458832}a^{7}-\frac{8882021}{91152944}a^{6}-\frac{1058788}{17091177}a^{5}+\frac{7652905}{45576472}a^{4}+\frac{21819457}{91152944}a^{3}-\frac{42657861}{91152944}a^{2}-\frac{89062615}{182305888}a+\frac{43901539}{182305888}$ 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

Trivial group, which has order $1$ (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:  $6$
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:   $\frac{7261}{2087472}a^{9}+\frac{267}{86978}a^{8}+\frac{10237}{521868}a^{7}+\frac{5981}{347912}a^{6}+\frac{55027}{1043736}a^{5}-\frac{261689}{347912}a^{4}-\frac{177043}{521868}a^{3}+\frac{627573}{347912}a^{2}-\frac{3174273}{695824}a+\frac{1679495}{347912}$, $\frac{550067}{273458832}a^{9}+\frac{40963}{22788236}a^{8}+\frac{269705}{34182354}a^{7}+\frac{49}{45576472}a^{6}-\frac{805619}{136729416}a^{5}-\frac{23833287}{45576472}a^{4}-\frac{4736157}{11394118}a^{3}+\frac{68735445}{45576472}a^{2}-\frac{107568871}{91152944}a+\frac{67021663}{45576472}$, $\frac{63757}{17091177}a^{9}+\frac{910547}{136729416}a^{8}+\frac{1257971}{45576472}a^{7}+\frac{4496455}{136729416}a^{6}+\frac{3410049}{45576472}a^{5}-\frac{109742893}{136729416}a^{4}-\frac{147990265}{136729416}a^{3}+\frac{32768091}{45576472}a^{2}-\frac{71016957}{45576472}a+\frac{118914817}{22788236}$, $\frac{3906391}{546917664}a^{9}+\frac{758921}{182305888}a^{8}+\frac{3730547}{91152944}a^{7}+\frac{3407955}{91152944}a^{6}+\frac{710552}{5697059}a^{5}-\frac{66492527}{45576472}a^{4}-\frac{7218679}{273458832}a^{3}+\frac{361030927}{91152944}a^{2}-\frac{2162730953}{182305888}a+\frac{2236976527}{182305888}$, $\frac{4595335}{546917664}a^{9}+\frac{750165}{182305888}a^{8}+\frac{7871773}{273458832}a^{7}-\frac{595969}{91152944}a^{6}+\frac{3219281}{68364708}a^{5}-\frac{89308819}{45576472}a^{4}-\frac{22625713}{91152944}a^{3}+\frac{810752499}{91152944}a^{2}-\frac{1667129441}{182305888}a+\frac{452359699}{182305888}$, $\frac{1809115}{22788236}a^{9}+\frac{93418841}{273458832}a^{8}+\frac{17308342}{17091177}a^{7}+\frac{286932881}{136729416}a^{6}+\frac{61702292}{17091177}a^{5}-\frac{881483413}{68364708}a^{4}-\frac{4395149345}{68364708}a^{3}-\frac{2531670971}{45576472}a^{2}+\frac{222524514}{5697059}a+\frac{8261987401}{91152944}$ 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:  \( 18035.1495626 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 18035.1495626 \cdot 1}{2\cdot\sqrt{3690562500000000}}\cr\approx \mathstrut & 0.589119038481 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359)
 
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 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359, 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 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359);
 
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 + 5*x^8 + 10*x^6 - 228*x^5 + 90*x^4 + 600*x^3 - 1755*x^2 + 2700*x - 1359);
 
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

$S_5\wr C_2$ (as 10T43):

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 28800
The 35 conjugacy class representatives for $S_5^2 \wr C_2$
Character table for $S_5^2 \wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 24 siblings: data not computed
Degree 25 sibling: data not computed
Degree 30 sibling: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: 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 R ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 2.4.4.4$x^{4} - 2 x^{3} + 4 x^{2} + 12 x + 12$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.4.2.1$x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.6.8.8$x^{6} + 9 x^{4} + 24 x^{3} + 144$$3$$2$$8$$S_3\times C_3$$[2, 2]^{2}$
\(5\) Copy content Toggle raw display 5.10.12.15$x^{10} + 5 x^{3} + 5$$10$$1$$12$$(C_5^2 : C_8):C_2$$[11/8, 11/8]_{8}^{2}$