Properties

Label 10.10.5000000000000000.1
Degree $10$
Signature $[10, 0]$
Discriminant $5.000\times 10^{15}$
Root discriminant \(37.14\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $C_{10}$ (as 10T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49)
 
gp: K = bnfinit(y^10 - 30*y^8 - 10*y^7 + 240*y^6 + 78*y^5 - 655*y^4 - 320*y^3 + 570*y^2 + 420*y + 49, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49)
 

\( x^{10} - 30x^{8} - 10x^{7} + 240x^{6} + 78x^{5} - 655x^{4} - 320x^{3} + 570x^{2} + 420x + 49 \) 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:  $[10, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(5000000000000000\) \(\medspace = 2^{15}\cdot 5^{16}\) 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:  \(37.14\)
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:  $2^{3/2}5^{8/5}\approx 37.14471242937835$
Ramified primes:   \(2\), \(5\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(200=2^{3}\cdot 5^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(101,·)$, $\chi_{200}(161,·)$, $\chi_{200}(41,·)$, $\chi_{200}(141,·)$, $\chi_{200}(81,·)$, $\chi_{200}(21,·)$, $\chi_{200}(121,·)$, $\chi_{200}(61,·)$, $\chi_{200}(181,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{359807}a^{9}-\frac{839}{359807}a^{8}-\frac{15723}{359807}a^{7}-\frac{18470}{359807}a^{6}-\frac{26532}{359807}a^{5}+\frac{106595}{359807}a^{4}+\frac{55281}{359807}a^{3}-\frac{17377}{359807}a^{2}-\frac{2573}{51401}a+\frac{3142}{7343}$ 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:  $7$

Class group and class number

Trivial group, which has order $1$

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:  $9$
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:   $\frac{17490}{359807}a^{9}-\frac{24825}{359807}a^{8}-\frac{513930}{359807}a^{7}+\frac{528995}{359807}a^{6}+\frac{4115628}{359807}a^{5}-\frac{3565790}{359807}a^{4}-\frac{10268320}{359807}a^{3}+\frac{4225265}{359807}a^{2}+\frac{1295845}{51401}a+\frac{27940}{7343}$, $\frac{3380}{51401}a^{9}-\frac{1422}{51401}a^{8}-\frac{97908}{51401}a^{7}+\frac{8929}{51401}a^{6}+\frac{728856}{51401}a^{5}-\frac{110683}{51401}a^{4}-\frac{1703774}{51401}a^{3}-\frac{49004}{51401}a^{2}+\frac{184094}{7343}a+\frac{7227}{1049}$, $\frac{24340}{359807}a^{9}-\frac{15063}{359807}a^{8}-\frac{685588}{359807}a^{7}+\frac{198950}{359807}a^{6}+\frac{4742176}{359807}a^{5}-\frac{1845012}{359807}a^{4}-\frac{8774008}{359807}a^{3}+\frac{1718882}{359807}a^{2}+\frac{611096}{51401}a+\frac{6278}{7343}$, $\frac{17490}{359807}a^{9}-\frac{24825}{359807}a^{8}-\frac{513930}{359807}a^{7}+\frac{528995}{359807}a^{6}+\frac{4115628}{359807}a^{5}-\frac{3565790}{359807}a^{4}-\frac{10268320}{359807}a^{3}+\frac{4225265}{359807}a^{2}+\frac{1244444}{51401}a+\frac{35283}{7343}$, $\frac{680}{359807}a^{9}-\frac{5109}{359807}a^{8}-\frac{232}{359807}a^{7}+\frac{136447}{359807}a^{6}-\frac{359816}{359807}a^{5}-\frac{1070231}{359807}a^{4}+\frac{3152410}{359807}a^{3}+\frac{2061910}{359807}a^{2}-\frac{677562}{51401}a-\frac{58997}{7343}$, $\frac{2589}{51401}a^{9}+\frac{8700}{51401}a^{8}-\frac{63342}{51401}a^{7}-\frac{250876}{51401}a^{6}+\frac{171306}{51401}a^{5}+\frac{1228767}{51401}a^{4}+\frac{352560}{51401}a^{3}-\frac{1540522}{51401}a^{2}-\frac{154550}{7343}a-\frac{2455}{1049}$, $\frac{33452}{359807}a^{9}-\frac{52683}{359807}a^{8}-\frac{955982}{359807}a^{7}+\frac{1163803}{359807}a^{6}+\frac{7188743}{359807}a^{5}-\frac{8198392}{359807}a^{4}-\frac{15465466}{359807}a^{3}+\frac{11255324}{359807}a^{2}+\frac{1603424}{51401}a+\frac{27854}{7343}$, $\frac{95336}{359807}a^{9}+\frac{95854}{359807}a^{8}-\frac{2582016}{359807}a^{7}-\frac{3455730}{359807}a^{6}+\frac{14432139}{359807}a^{5}+\frac{17788159}{359807}a^{4}-\frac{14832805}{359807}a^{3}-\frac{25802744}{359807}a^{2}-\frac{1467870}{51401}a-\frac{26659}{7343}$, $\frac{6936}{359807}a^{9}-\frac{10991}{359807}a^{8}-\frac{187410}{359807}a^{7}+\frac{240377}{359807}a^{6}+\frac{1223252}{359807}a^{5}-\frac{1705297}{359807}a^{4}-\frac{2027083}{359807}a^{3}+\frac{1344899}{359807}a^{2}+\frac{290882}{51401}a+\frac{13574}{7343}$ Copy content Toggle raw display
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:  \( 82605.472142 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 82605.472142 \cdot 1}{2\cdot\sqrt{5000000000000000}}\cr\approx \mathstrut & 0.59812750863 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49)
 
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 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49, 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 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49);
 
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 - 30*x^8 - 10*x^7 + 240*x^6 + 78*x^5 - 655*x^4 - 320*x^3 + 570*x^2 + 420*x + 49);
 
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

$C_{10}$ (as 10T1):

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 cyclic group of order 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.390625.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }$ R ${\href{/padicField/7.1.0.1}{1} }^{10}$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.10.0.1}{10} }$ ${\href{/padicField/17.5.0.1}{5} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}$ ${\href{/padicField/29.10.0.1}{10} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

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 2.10.15.1$x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$$2$$5$$15$$C_{10}$$[3]^{5}$
\(5\) Copy content Toggle raw display 5.10.16.7$x^{10} + 40 x^{9} + 400 x^{8} + 10 x^{5} + 200 x^{4} - 1225$$5$$2$$16$$C_{10}$$[2]^{2}$