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Properties

Label 20.0.369...048.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.691\times 10^{20}$
Root discriminant \(10.67\)
Ramified primes $2,3,157$
Class number $1$
Class group trivial
Galois group $D_5\wr C_2$ (as 20T48)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1)
 
gp: K = bnfinit(y^20 - 7*y^19 + 26*y^18 - 64*y^17 + 116*y^16 - 153*y^15 + 128*y^14 - 4*y^13 - 199*y^12 + 398*y^11 - 478*y^10 + 372*y^9 - 118*y^8 - 158*y^7 + 328*y^6 - 344*y^5 + 249*y^4 - 129*y^3 + 46*y^2 - 10*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1)
 

\( x^{20} - 7 x^{19} + 26 x^{18} - 64 x^{17} + 116 x^{16} - 153 x^{15} + 128 x^{14} - 4 x^{13} - 199 x^{12} + \cdots + 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(369139687194512130048\) \(\medspace = 2^{16}\cdot 3^{10}\cdot 157^{5}\) 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:  \(10.67\)
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:  $2^{4/5}3^{1/2}157^{1/2}\approx 37.7863071184893$
Ramified primes:   \(2\), \(3\), \(157\) 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:  \(\Q(\sqrt{157}) \)
$\card{ \Aut(K/\Q) }$:  $10$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31}a^{18}-\frac{1}{31}a^{17}-\frac{5}{31}a^{16}-\frac{7}{31}a^{15}+\frac{13}{31}a^{14}+\frac{7}{31}a^{13}+\frac{7}{31}a^{11}-\frac{2}{31}a^{10}-\frac{6}{31}a^{9}+\frac{1}{31}a^{8}+\frac{1}{31}a^{7}-\frac{13}{31}a^{6}-\frac{13}{31}a^{5}-\frac{14}{31}a^{4}-\frac{10}{31}a^{3}+\frac{12}{31}a^{2}+\frac{7}{31}a+\frac{5}{31}$, $\frac{1}{5053}a^{19}-\frac{54}{5053}a^{18}-\frac{696}{5053}a^{17}+\frac{537}{5053}a^{16}+\frac{1283}{5053}a^{15}-\frac{73}{163}a^{14}-\frac{1580}{5053}a^{13}+\frac{906}{5053}a^{12}-\frac{2357}{5053}a^{11}+\frac{2456}{5053}a^{10}+\frac{1776}{5053}a^{9}+\frac{2149}{5053}a^{8}+\frac{1732}{5053}a^{7}+\frac{1079}{5053}a^{6}+\frac{2101}{5053}a^{5}-\frac{1128}{5053}a^{4}-\frac{2155}{5053}a^{3}+\frac{2378}{5053}a^{2}+\frac{750}{5053}a-\frac{2497}{5053}$ 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

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:   \( \frac{92}{31} a^{19} - \frac{627}{31} a^{18} + \frac{2307}{31} a^{17} - \frac{5626}{31} a^{16} + \frac{10118}{31} a^{15} - \frac{13131}{31} a^{14} + \frac{10639}{31} a^{13} + \frac{551}{31} a^{12} - \frac{18282}{31} a^{11} + \frac{35176}{31} a^{10} - \frac{40966}{31} a^{9} + \frac{30588}{31} a^{8} - \frac{7714}{31} a^{7} - \frac{15910}{31} a^{6} + \frac{29382}{31} a^{5} - \frac{29235}{31} a^{4} + \frac{20032}{31} a^{3} - \frac{9527}{31} a^{2} + \frac{2853}{31} a - \frac{381}{31} \)  (order $6$) 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:   $\frac{25329}{5053}a^{19}-\frac{152764}{5053}a^{18}+\frac{503951}{5053}a^{17}-\frac{1093755}{5053}a^{16}+\frac{1753840}{5053}a^{15}-\frac{1919087}{5053}a^{14}+\frac{1001249}{5053}a^{13}+\frac{1225227}{5053}a^{12}-\frac{3919571}{5053}a^{11}+\frac{5806927}{5053}a^{10}-\frac{5478540}{5053}a^{9}+\frac{2903809}{5053}a^{8}+\frac{699278}{5053}a^{7}-\frac{3477498}{5053}a^{6}+\frac{4379636}{5053}a^{5}-\frac{3575551}{5053}a^{4}+\frac{2047421}{5053}a^{3}-\frac{820918}{5053}a^{2}+\frac{210511}{5053}a-\frac{32179}{5053}$, $\frac{3022}{5053}a^{19}-\frac{8827}{5053}a^{18}+\frac{1018}{5053}a^{17}+\frac{67794}{5053}a^{16}-\frac{224995}{5053}a^{15}+\frac{463451}{5053}a^{14}-\frac{632115}{5053}a^{13}+\frac{514612}{5053}a^{12}+\frac{61697}{5053}a^{11}-\frac{905801}{5053}a^{10}+\frac{1641544}{5053}a^{9}-\frac{1759553}{5053}a^{8}+\frac{1108574}{5053}a^{7}-\frac{9205}{5053}a^{6}-\frac{932803}{5053}a^{5}+\frac{1297157}{5053}a^{4}-\frac{1092986}{5053}a^{3}+\frac{620350}{5053}a^{2}-\frac{220391}{5053}a+\frac{37315}{5053}$, $\frac{661}{163}a^{19}-\frac{124928}{5053}a^{18}+\frac{420978}{5053}a^{17}-\frac{938039}{5053}a^{16}+\frac{1551443}{5053}a^{15}-\frac{1776615}{5053}a^{14}+\frac{1074102}{5053}a^{13}+\frac{27388}{163}a^{12}-\frac{3336671}{5053}a^{11}+\frac{5260634}{5053}a^{10}-\frac{5227263}{5053}a^{9}+\frac{3066739}{5053}a^{8}+\frac{287465}{5053}a^{7}-\frac{3055944}{5053}a^{6}+\frac{4126539}{5053}a^{5}-\frac{3480987}{5053}a^{4}+\frac{2043593}{5053}a^{3}-\frac{811525}{5053}a^{2}+\frac{183007}{5053}a-\frac{13041}{5053}$, $\frac{9688}{5053}a^{19}-\frac{62514}{5053}a^{18}+\frac{214318}{5053}a^{17}-\frac{481191}{5053}a^{16}+\frac{786940}{5053}a^{15}-\frac{887816}{5053}a^{14}+\frac{489290}{5053}a^{13}+\frac{520726}{5053}a^{12}-\frac{1788219}{5053}a^{11}+\frac{2670558}{5053}a^{10}-\frac{2541073}{5053}a^{9}+\frac{1315747}{5053}a^{8}+\frac{383446}{5053}a^{7}-\frac{1679390}{5053}a^{6}+\frac{2046980}{5053}a^{5}-\frac{1616689}{5053}a^{4}+\frac{867375}{5053}a^{3}-\frac{297016}{5053}a^{2}+\frac{50968}{5053}a+\frac{1850}{5053}$, $\frac{8673}{5053}a^{19}-\frac{57093}{5053}a^{18}+\frac{207144}{5053}a^{17}-\frac{501386}{5053}a^{16}+\frac{901654}{5053}a^{15}-\frac{1168227}{5053}a^{14}+\frac{948893}{5053}a^{13}+\frac{45800}{5053}a^{12}-\frac{1626356}{5053}a^{11}+\frac{3146600}{5053}a^{10}-\frac{3668404}{5053}a^{9}+\frac{2743495}{5053}a^{8}-\frac{22298}{163}a^{7}-\frac{1440257}{5053}a^{6}+\frac{2648464}{5053}a^{5}-\frac{2625162}{5053}a^{4}+\frac{1785093}{5053}a^{3}-\frac{837490}{5053}a^{2}+\frac{247669}{5053}a-\frac{29967}{5053}$, $\frac{17}{31}a^{19}-\frac{85}{31}a^{18}+\frac{262}{31}a^{17}-\frac{554}{31}a^{16}+\frac{945}{31}a^{15}-\frac{1137}{31}a^{14}+\frac{919}{31}a^{13}+\frac{26}{31}a^{12}-\frac{1440}{31}a^{11}+\frac{3010}{31}a^{10}-\frac{3636}{31}a^{9}+\frac{3142}{31}a^{8}-\frac{1374}{31}a^{7}-\frac{794}{31}a^{6}+\frac{2413}{31}a^{5}-\frac{2876}{31}a^{4}+\frac{2341}{31}a^{3}-\frac{1379}{31}a^{2}+\frac{508}{31}a-\frac{92}{31}$, $\frac{21149}{5053}a^{19}-\frac{145627}{5053}a^{18}+\frac{529272}{5053}a^{17}-\frac{1270271}{5053}a^{16}+\frac{2236243}{5053}a^{15}-\frac{2830296}{5053}a^{14}+\frac{2149387}{5053}a^{13}+\frac{409311}{5053}a^{12}-\frac{4278446}{5053}a^{11}+\frac{7746450}{5053}a^{10}-\frac{8690300}{5053}a^{9}+\frac{6097415}{5053}a^{8}-\frac{1074440}{5053}a^{7}-\frac{3827306}{5053}a^{6}+\frac{6377298}{5053}a^{5}-\frac{6047833}{5053}a^{4}+\frac{3968896}{5053}a^{3}-\frac{1797425}{5053}a^{2}+\frac{522635}{5053}a-\frac{71055}{5053}$, $\frac{13699}{5053}a^{19}-\frac{91495}{5053}a^{18}+\frac{322432}{5053}a^{17}-\frac{745878}{5053}a^{16}+\frac{1259517}{5053}a^{15}-\frac{1497511}{5053}a^{14}+\frac{967918}{5053}a^{13}+\frac{582221}{5053}a^{12}-\frac{2687906}{5053}a^{11}+\frac{4339463}{5053}a^{10}-\frac{4436001}{5053}a^{9}+\frac{2654665}{5053}a^{8}+\frac{165996}{5053}a^{7}-\frac{2544375}{5053}a^{6}+\frac{3472263}{5053}a^{5}-\frac{2926411}{5053}a^{4}+\frac{1691525}{5053}a^{3}-\frac{629649}{5053}a^{2}+\frac{127989}{5053}a-\frac{5417}{5053}$, $\frac{17220}{5053}a^{19}-\frac{116510}{5053}a^{18}+\frac{420158}{5053}a^{17}-\frac{999529}{5053}a^{16}+\frac{1745970}{5053}a^{15}-\frac{2180086}{5053}a^{14}+\frac{1613571}{5053}a^{13}+\frac{412002}{5053}a^{12}-\frac{3398601}{5053}a^{11}+\frac{6027265}{5053}a^{10}-\frac{6641753}{5053}a^{9}+\frac{4570410}{5053}a^{8}-\frac{669978}{5053}a^{7}-\frac{3029182}{5053}a^{6}+\frac{4913375}{5053}a^{5}-\frac{4596376}{5053}a^{4}+\frac{2993138}{5053}a^{3}-\frac{1351459}{5053}a^{2}+\frac{392525}{5053}a-\frac{53708}{5053}$ 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:  \( 182.345344476 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 182.345344476 \cdot 1}{6\cdot\sqrt{369139687194512130048}}\cr\approx \mathstrut & 0.151686434459 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1)
 
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^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1, 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^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1);
 
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^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1);
 
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

$D_5\wr C_2$ (as 20T48):

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 solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.1413.1, 10.0.1533364992.1 x5

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 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.0.370872403061616.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.4.0.1}{4} }^{5}$ ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{5}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.4.0.1}{4} }^{5}$ ${\href{/padicField/29.4.0.1}{4} }^{5}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.2.0.1}{2} }^{10}$ ${\href{/padicField/41.4.0.1}{4} }^{5}$ ${\href{/padicField/43.5.0.1}{5} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{5}$ ${\href{/padicField/59.4.0.1}{4} }^{5}$

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 Deg $20$$5$$4$$16$
\(3\) Copy content Toggle raw display 3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(157\) Copy content Toggle raw display 157.2.1.1$x^{2} + 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} + 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} + 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} + 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.2.1.1$x^{2} + 157$$2$$1$$1$$C_2$$[\ ]_{2}$
157.5.0.1$x^{5} + 7 x + 152$$1$$5$$0$$C_5$$[\ ]^{5}$
157.5.0.1$x^{5} + 7 x + 152$$1$$5$$0$$C_5$$[\ ]^{5}$