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Properties

Label 10.2.204...000.26
Degree $10$
Signature $[2, 4]$
Discriminant $2.048\times 10^{21}$
Root discriminant \(135.25\)
Ramified primes $2,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(A_5^2 : C_2):C_2$ (as 10T41)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800)
 
Copy content gp:K = bnfinit(y^10 - 100*y^8 - 200*y^7 + 4550*y^6 + 18920*y^5 - 92500*y^4 - 651000*y^3 + 78625*y^2 + 7927000*y + 15270800, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800)
 

\( x^{10} - 100 x^{8} - 200 x^{7} + 4550 x^{6} + 18920 x^{5} - 92500 x^{4} - 651000 x^{3} + 78625 x^{2} + \cdots + 15270800 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $10$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 4]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(2048000000000000000000\) \(\medspace = 2^{29}\cdot 5^{18}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(135.25\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{55/16}5^{203/100}\approx 284.2495061971761$
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{40}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{80}a^{6}-\frac{1}{80}a^{5}-\frac{1}{8}a^{4}-\frac{1}{8}a^{3}-\frac{7}{16}a^{2}+\frac{7}{16}a+\frac{1}{4}$, $\frac{1}{80}a^{7}-\frac{1}{80}a^{5}-\frac{1}{16}a^{3}+\frac{1}{4}a^{2}+\frac{1}{16}a-\frac{1}{4}$, $\frac{1}{160}a^{8}-\frac{1}{160}a^{6}-\frac{1}{80}a^{5}+\frac{3}{32}a^{4}+\frac{1}{8}a^{3}-\frac{11}{32}a^{2}+\frac{3}{16}a-\frac{1}{4}$, $\frac{1}{351343445353920}a^{9}-\frac{780397830877}{351343445353920}a^{8}+\frac{560560274519}{117114481784640}a^{7}+\frac{760472999623}{351343445353920}a^{6}-\frac{364709213549}{351343445353920}a^{5}+\frac{3029248633469}{70268689070784}a^{4}+\frac{16029755006627}{70268689070784}a^{3}-\frac{489984475055}{70268689070784}a^{2}-\frac{3407458879021}{8783586133848}a+\frac{707826756061}{4391793066924}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $5$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{125}{312811344}a^{9}-\frac{475}{156405672}a^{8}-\frac{110}{6516903}a^{7}+\frac{1891}{39101418}a^{6}+\frac{1134443}{782028360}a^{5}+\frac{240175}{156405672}a^{4}-\frac{1901645}{39101418}a^{3}-\frac{10989455}{78202836}a^{2}+\frac{187501885}{312811344}a+\frac{214051867}{78202836}$, $\frac{1756396441829}{351343445353920}a^{9}-\frac{10794725722297}{351343445353920}a^{8}+\frac{40196064469757}{117114481784640}a^{7}+\frac{12\cdots 63}{351343445353920}a^{6}-\frac{44660151813883}{70268689070784}a^{5}-\frac{82\cdots 71}{70268689070784}a^{4}-\frac{26\cdots 27}{70268689070784}a^{3}+\frac{59\cdots 05}{70268689070784}a^{2}+\frac{30\cdots 83}{4391793066924}a+\frac{47\cdots 27}{4391793066924}$, $\frac{75\cdots 31}{70268689070784}a^{9}-\frac{14\cdots 71}{351343445353920}a^{8}-\frac{10\cdots 83}{117114481784640}a^{7}+\frac{47\cdots 53}{351343445353920}a^{6}+\frac{30\cdots 61}{70268689070784}a^{5}+\frac{26\cdots 19}{70268689070784}a^{4}-\frac{79\cdots 71}{70268689070784}a^{3}-\frac{18\cdots 93}{70268689070784}a^{2}+\frac{24\cdots 59}{2195896533462}a+\frac{19\cdots 53}{4391793066924}$, $\frac{14901802647005}{17567172267696}a^{9}+\frac{32\cdots 09}{175671722676960}a^{8}+\frac{48\cdots 59}{14639310223080}a^{7}+\frac{91\cdots 19}{175671722676960}a^{6}-\frac{14\cdots 83}{87835861338480}a^{5}-\frac{37\cdots 73}{35134344535392}a^{4}+\frac{35\cdots 85}{4391793066924}a^{3}+\frac{80\cdots 53}{35134344535392}a^{2}+\frac{77\cdots 83}{8783586133848}a+\frac{12\cdots 77}{1097948266731}$, $\frac{90\cdots 11}{351343445353920}a^{9}-\frac{68\cdots 51}{70268689070784}a^{8}-\frac{25\cdots 11}{117114481784640}a^{7}+\frac{11\cdots 21}{351343445353920}a^{6}+\frac{73\cdots 77}{70268689070784}a^{5}+\frac{62\cdots 59}{70268689070784}a^{4}-\frac{19\cdots 47}{70268689070784}a^{3}-\frac{45\cdots 45}{70268689070784}a^{2}+\frac{46\cdots 89}{17567172267696}a+\frac{11\cdots 24}{1097948266731}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 14492543.076974966 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 14492543.076974966 \cdot 1}{2\cdot\sqrt{2048000000000000000000}}\cr\approx \mathstrut & 0.998226495247556 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 100*x^8 - 200*x^7 + 4550*x^6 + 18920*x^5 - 92500*x^4 - 651000*x^3 + 78625*x^2 + 7927000*x + 15270800); 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

$\POPlus(4,5)$ (as 10T41):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 14400
The 25 conjugacy class representatives for $(A_5^2 : C_2):C_2$
Character table for $(A_5^2 : C_2):C_2$

Intermediate fields

\(\Q(\sqrt{2}) \)

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(b)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: deg 12
Degree 20 siblings: deg 20, deg 20
Degree 24 sibling: deg 24
Degree 25 sibling: deg 25
Degree 30 sibling: deg 30
Degree 36 sibling: deg 36
Degree 40 siblings: deg 40, deg 40, deg 40
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 ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.1.8.26c1.2$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 2$$8$$1$$26$$(((C_4 \times C_2): C_2):C_2):C_2$$$[2, 2, 3, \frac{7}{2}, 4]^{2}$$
\(5\) Copy content Toggle raw display 5.2.5.18a1.15$x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2280 x^{6} + 4064 x^{5} + 4560 x^{4} + 3250 x^{3} + 1660 x^{2} + 820 x + 237$$5$$2$$18$$(C_5^2 : C_4) : C_2$$$[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)