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Properties

Label 10.0.294...875.1
Degree $10$
Signature $[0, 5]$
Discriminant $-2.942\times 10^{26}$
Root discriminant \(443.47\)
Ramified primes $5,11,41$
Class number $25$ (GRH)
Class group [5, 5] (GRH)
Galois group $F_{5}\times C_2$ (as 10T5)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265)
 
Copy content gp:K = bnfinit(y^10 - y^9 + 375*y^8 - 1310*y^7 + 40085*y^6 - 207703*y^5 + 1399773*y^4 - 4540790*y^3 + 10921175*y^2 - 13775785*y + 11330265, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265)
 

\( x^{10} - x^{9} + 375 x^{8} - 1310 x^{7} + 40085 x^{6} - 207703 x^{5} + 1399773 x^{4} - 4540790 x^{3} + \cdots + 11330265 \) 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:  $[0, 5]$
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:   \(-294188062606461061087671875\) \(\medspace = -\,5^{6}\cdot 11^{9}\cdot 41^{8}\) 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:  \(443.47\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{3/4}11^{9/10}41^{4/5}\approx 564.557271214292$
Ramified primes:   \(5\), \(11\), \(41\) 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:  \(\Q(\sqrt{-11}) \)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-11}) \)

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{8}a^{4}-\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{88}a^{5}+\frac{5}{88}a^{4}-\frac{1}{88}a^{3}+\frac{21}{88}a^{2}-\frac{39}{88}a-\frac{21}{88}$, $\frac{1}{352}a^{6}+\frac{1}{352}a^{5}-\frac{5}{176}a^{4}-\frac{63}{352}a^{3}-\frac{23}{176}a^{2}-\frac{107}{352}a+\frac{161}{352}$, $\frac{1}{352}a^{7}+\frac{1}{352}a^{5}+\frac{7}{352}a^{4}+\frac{5}{352}a^{3}+\frac{15}{352}a^{2}-\frac{3}{44}a+\frac{115}{352}$, $\frac{1}{1408}a^{8}-\frac{1}{704}a^{7}-\frac{1}{704}a^{6}-\frac{3}{704}a^{5}-\frac{63}{1408}a^{4}+\frac{13}{704}a^{3}+\frac{17}{176}a^{2}-\frac{43}{352}a-\frac{149}{1408}$, $\frac{1}{21\cdots 40}a^{9}+\frac{664580401527}{21\cdots 40}a^{8}-\frac{247696510591}{535229137996960}a^{7}-\frac{233147578549}{267614568998480}a^{6}+\frac{5382865833779}{21\cdots 40}a^{5}+\frac{133349457205739}{21\cdots 40}a^{4}+\frac{8357401714457}{214091655198784}a^{3}+\frac{4315106962401}{107045827599392}a^{2}+\frac{12678291901059}{428183310397568}a-\frac{143096963940449}{428183310397568}$ 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:  $C_{5}\times C_{5}$, which has order $25$ (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:  $C_{5}\times C_{5}$, which has order $25$ (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:  $4$
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:   $\frac{99\cdots 37}{194628777453440}a^{9}-\frac{31\cdots 93}{24328597181680}a^{8}-\frac{85\cdots 71}{97314388726720}a^{7}-\frac{30\cdots 73}{97314388726720}a^{6}+\frac{35\cdots 07}{194628777453440}a^{5}-\frac{22\cdots 91}{1520537323855}a^{4}+\frac{50\cdots 71}{9731438872672}a^{3}-\frac{31\cdots 85}{2432859718168}a^{2}+\frac{58\cdots 79}{3538705044608}a-\frac{26\cdots 43}{19462877745344}$, $\frac{20\cdots 31}{9731438872672}a^{9}+\frac{19\cdots 89}{1769352522304}a^{8}-\frac{38\cdots 15}{4865719436336}a^{7}+\frac{11\cdots 23}{4865719436336}a^{6}-\frac{39\cdots 57}{4865719436336}a^{5}+\frac{77\cdots 05}{19462877745344}a^{4}-\frac{24\cdots 23}{9731438872672}a^{3}+\frac{67\cdots 45}{884676261152}a^{2}-\frac{33\cdots 55}{2432859718168}a+\frac{21\cdots 89}{19462877745344}$, $\frac{20\cdots 27}{214091655198784}a^{9}+\frac{34\cdots 35}{214091655198784}a^{8}+\frac{36\cdots 59}{107045827599392}a^{7}-\frac{48\cdots 25}{53522913799696}a^{6}+\frac{63\cdots 75}{214091655198784}a^{5}-\frac{32\cdots 67}{214091655198784}a^{4}+\frac{13\cdots 27}{26761456899848}a^{3}-\frac{90\cdots 25}{107045827599392}a^{2}+\frac{19\cdots 21}{214091655198784}a-\frac{89\cdots 31}{214091655198784}$, $\frac{16\cdots 79}{66903642249620}a^{9}-\frac{35\cdots 23}{10\cdots 20}a^{8}-\frac{39\cdots 66}{16725910562405}a^{7}-\frac{78\cdots 43}{16725910562405}a^{6}+\frac{12\cdots 67}{535229137996960}a^{5}-\frac{23\cdots 21}{10\cdots 20}a^{4}+\frac{81\cdots 63}{107045827599392}a^{3}-\frac{20\cdots 79}{107045827599392}a^{2}+\frac{27\cdots 71}{107045827599392}a-\frac{46\cdots 85}{214091655198784}$ 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:  \( 211938177.9326882 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 211938177.9326882 \cdot 25}{2\cdot\sqrt{294188062606461061087671875}}\cr\approx \mathstrut & 1.51253707228861 \end{aligned}\] (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 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265) 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 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265, 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 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265); 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 = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 + 375*x^8 - 1310*x^7 + 40085*x^6 - 207703*x^5 + 1399773*x^4 - 4540790*x^3 + 10921175*x^2 - 13775785*x + 11330265); 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

$C_2\times F_5$ (as 10T5):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 40
The 10 conjugacy class representatives for $F_{5}\times C_2$
Character table for $F_{5}\times C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 5.1.5171495850125.3

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Galois closure: data not computed
Degree 10 sibling: data not computed
Degree 20 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 ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }$ ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ R ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.2.0.1}{2} }^{5}$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{5}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ R ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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

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
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
\(11\) Copy content Toggle raw display 11.1.10.9a1.4$x^{10} + 44$$10$$1$$9$$C_{10}$$$[\ ]_{10}$$
\(41\) Copy content Toggle raw display 41.2.5.8a1.1$x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$$5$$2$$8$$C_{10}$$$[\ ]_{5}^{2}$$

Spectrum of ring of integers

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