Properties

Label 20.0.56946837901...6801.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 17^{10}$
Root discriminant $10.91$
Ramified primes $7, 17$
Class number $1$
Class group Trivial
Galois Group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -12, 9, 7, 30, -58, 12, 4, 41, -63, 37, -4, -13, 11, 6, -12, 8, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 8*x^18 - 12*x^17 + 6*x^16 + 11*x^15 - 13*x^14 - 4*x^13 + 37*x^12 - 63*x^11 + 41*x^10 + 4*x^9 + 12*x^8 - 58*x^7 + 30*x^6 + 7*x^5 + 9*x^4 - 12*x^3 + 1)
gp: K = bnfinit(x^20 - 4*x^19 + 8*x^18 - 12*x^17 + 6*x^16 + 11*x^15 - 13*x^14 - 4*x^13 + 37*x^12 - 63*x^11 + 41*x^10 + 4*x^9 + 12*x^8 - 58*x^7 + 30*x^6 + 7*x^5 + 9*x^4 - 12*x^3 + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 4 x^{19} \) \(\mathstrut +\mathstrut 8 x^{18} \) \(\mathstrut -\mathstrut 12 x^{17} \) \(\mathstrut +\mathstrut 6 x^{16} \) \(\mathstrut +\mathstrut 11 x^{15} \) \(\mathstrut -\mathstrut 13 x^{14} \) \(\mathstrut -\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 37 x^{12} \) \(\mathstrut -\mathstrut 63 x^{11} \) \(\mathstrut +\mathstrut 41 x^{10} \) \(\mathstrut +\mathstrut 4 x^{9} \) \(\mathstrut +\mathstrut 12 x^{8} \) \(\mathstrut -\mathstrut 58 x^{7} \) \(\mathstrut +\mathstrut 30 x^{6} \) \(\mathstrut +\mathstrut 7 x^{5} \) \(\mathstrut +\mathstrut 9 x^{4} \) \(\mathstrut -\mathstrut 12 x^{3} \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(569468379011812486801=7^{10}\cdot 17^{10}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.91$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $7, 17$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{18} - \frac{1}{26} a^{17} + \frac{1}{26} a^{16} - \frac{5}{26} a^{15} - \frac{1}{2} a^{14} + \frac{5}{26} a^{13} + \frac{1}{13} a^{12} - \frac{5}{26} a^{11} - \frac{6}{13} a^{10} + \frac{6}{13} a^{9} - \frac{5}{26} a^{8} - \frac{7}{26} a^{7} - \frac{1}{13} a^{6} + \frac{3}{26} a^{5} + \frac{4}{13} a^{4} - \frac{7}{26} a^{3} - \frac{5}{26} a^{2} - \frac{6}{13} a + \frac{5}{13}$, $\frac{1}{642049175534} a^{19} - \frac{3783435379}{321024587767} a^{18} - \frac{67684022219}{642049175534} a^{17} + \frac{146314195525}{642049175534} a^{16} - \frac{63574974167}{321024587767} a^{15} + \frac{5717541913}{642049175534} a^{14} + \frac{26458292713}{642049175534} a^{13} + \frac{262972886}{24694199059} a^{12} + \frac{98609335257}{321024587767} a^{11} - \frac{202713022049}{642049175534} a^{10} - \frac{45179684889}{642049175534} a^{9} - \frac{11646478351}{49388398118} a^{8} - \frac{15444796061}{49388398118} a^{7} + \frac{18353610819}{642049175534} a^{6} + \frac{37673426969}{642049175534} a^{5} + \frac{13774315511}{49388398118} a^{4} - \frac{226202287847}{642049175534} a^{3} - \frac{199286226}{321024587767} a^{2} + \frac{262217054027}{642049175534} a + \frac{58351713835}{642049175534}$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $9$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{240166146645}{321024587767} a^{19} - \frac{1310445850619}{642049175534} a^{18} + \frac{905969997580}{321024587767} a^{17} - \frac{2372405192537}{642049175534} a^{16} - \frac{828883779090}{321024587767} a^{15} + \frac{4835509356209}{642049175534} a^{14} + \frac{886113034126}{321024587767} a^{13} - \frac{2638948898076}{321024587767} a^{12} + \frac{11641001075313}{642049175534} a^{11} - \frac{9532894816987}{642049175534} a^{10} - \frac{5168211599455}{642049175534} a^{9} + \frac{2698819948018}{321024587767} a^{8} + \frac{19503138522431}{642049175534} a^{7} - \frac{18026665890067}{642049175534} a^{6} - \frac{7236632243164}{321024587767} a^{5} + \frac{1602439131604}{321024587767} a^{4} + \frac{12418663070739}{642049175534} a^{3} + \frac{719132017909}{321024587767} a^{2} - \frac{1187515946802}{321024587767} a - \frac{616919262187}{321024587767} \),  \( \frac{174602110903}{642049175534} a^{19} - \frac{31408350923}{49388398118} a^{18} + \frac{8103954368}{24694199059} a^{17} + \frac{296906080423}{642049175534} a^{16} - \frac{2507087483155}{642049175534} a^{15} + \frac{3693401649335}{642049175534} a^{14} + \frac{596821018160}{321024587767} a^{13} - \frac{4994003015895}{642049175534} a^{12} + \frac{5266697591575}{642049175534} a^{11} + \frac{29864352613}{49388398118} a^{10} - \frac{6051100909825}{321024587767} a^{9} + \frac{12907157845191}{642049175534} a^{8} + \frac{4677999391789}{642049175534} a^{7} - \frac{8407216810201}{642049175534} a^{6} - \frac{12075317110881}{642049175534} a^{5} + \frac{5897722688266}{321024587767} a^{4} + \frac{2177504154637}{321024587767} a^{3} - \frac{1171802417483}{642049175534} a^{2} - \frac{1469984750519}{321024587767} a + \frac{46599196231}{642049175534} \),  \( \frac{54438351297}{321024587767} a^{19} + \frac{33767409287}{642049175534} a^{18} - \frac{576443183811}{642049175534} a^{17} + \frac{44938450678}{24694199059} a^{16} - \frac{2885517637161}{642049175534} a^{15} + \frac{530669696569}{321024587767} a^{14} + \frac{3289654071779}{642049175534} a^{13} - \frac{2359401254277}{642049175534} a^{12} + \frac{1426613726363}{642049175534} a^{11} + \frac{3686023503299}{321024587767} a^{10} - \frac{12623052434365}{642049175534} a^{9} + \frac{3068078468035}{321024587767} a^{8} + \frac{2764789387318}{321024587767} a^{7} + \frac{1735496935098}{321024587767} a^{6} - \frac{6055033517431}{321024587767} a^{5} - \frac{975112027027}{642049175534} a^{4} + \frac{1427745898933}{642049175534} a^{3} + \frac{3407823759113}{642049175534} a^{2} - \frac{211216026365}{642049175534} a + \frac{38190195124}{321024587767} \),  \( \frac{642805205043}{642049175534} a^{19} - \frac{843944072255}{321024587767} a^{18} + \frac{1121761160583}{321024587767} a^{17} - \frac{216157108217}{49388398118} a^{16} - \frac{1433263178028}{321024587767} a^{15} + \frac{6722775107755}{642049175534} a^{14} + \frac{2479901328919}{642049175534} a^{13} - \frac{3691249645516}{321024587767} a^{12} + \frac{16447533831471}{642049175534} a^{11} - \frac{5837585071823}{321024587767} a^{10} - \frac{4586728046451}{321024587767} a^{9} + \frac{10318848448617}{642049175534} a^{8} + \frac{11694687534349}{321024587767} a^{7} - \frac{22064003944899}{642049175534} a^{6} - \frac{9236579704776}{321024587767} a^{5} + \frac{1382380734732}{321024587767} a^{4} + \frac{7633144784947}{321024587767} a^{3} + \frac{1416829047571}{642049175534} a^{2} - \frac{1917703550111}{642049175534} a - \frac{805459883112}{321024587767} \),  \( \frac{388375555935}{642049175534} a^{19} - \frac{1400779552131}{642049175534} a^{18} + \frac{2791586797933}{642049175534} a^{17} - \frac{2146963680334}{321024587767} a^{16} + \frac{877904206550}{321024587767} a^{15} + \frac{1813239399583}{321024587767} a^{14} - \frac{4479219895405}{642049175534} a^{13} + \frac{102556966217}{321024587767} a^{12} + \frac{13428932086613}{642049175534} a^{11} - \frac{11080887260702}{321024587767} a^{10} + \frac{7756692744635}{321024587767} a^{9} - \frac{320053223133}{642049175534} a^{8} + \frac{1330830710156}{321024587767} a^{7} - \frac{6527942121838}{321024587767} a^{6} + \frac{685954736915}{49388398118} a^{5} - \frac{4245265291235}{642049175534} a^{4} + \frac{45504261115}{321024587767} a^{3} - \frac{91508314675}{321024587767} a^{2} + \frac{1606034976841}{642049175534} a + \frac{243435181377}{642049175534} \),  \( \frac{75756725955}{321024587767} a^{19} - \frac{174899740153}{321024587767} a^{18} + \frac{69801513387}{321024587767} a^{17} + \frac{336068131037}{642049175534} a^{16} - \frac{2342384489947}{642049175534} a^{15} + \frac{3480671375685}{642049175534} a^{14} + \frac{972935214679}{642049175534} a^{13} - \frac{4131641506043}{642049175534} a^{12} + \frac{4608018222269}{642049175534} a^{11} - \frac{90798323052}{321024587767} a^{10} - \frac{11083961482785}{642049175534} a^{9} + \frac{5944488512926}{321024587767} a^{8} + \frac{1091085413972}{321024587767} a^{7} - \frac{5566301406255}{642049175534} a^{6} - \frac{888106823679}{49388398118} a^{5} + \frac{4204247806943}{321024587767} a^{4} + \frac{3867755106719}{642049175534} a^{3} + \frac{661786603111}{321024587767} a^{2} - \frac{2537584095807}{642049175534} a - \frac{217525697789}{642049175534} \),  \( \frac{41744790125}{321024587767} a^{19} - \frac{377035268269}{642049175534} a^{18} + \frac{728304363121}{642049175534} a^{17} - \frac{521830826973}{321024587767} a^{16} + \frac{523424383503}{642049175534} a^{15} + \frac{691575272747}{321024587767} a^{14} - \frac{1400397696099}{642049175534} a^{13} - \frac{598021975591}{642049175534} a^{12} + \frac{3283575132883}{642049175534} a^{11} - \frac{3165708060390}{321024587767} a^{10} + \frac{3229059273163}{642049175534} a^{9} + \frac{706793254399}{321024587767} a^{8} - \frac{73404853998}{321024587767} a^{7} - \frac{2804229513024}{321024587767} a^{6} + \frac{55873678925}{24694199059} a^{5} + \frac{1809511192017}{642049175534} a^{4} + \frac{1988171147147}{642049175534} a^{3} - \frac{777269184569}{642049175534} a^{2} - \frac{1151164985483}{642049175534} a - \frac{4670237333}{321024587767} \),  \( \frac{212937288145}{642049175534} a^{19} - \frac{608510130673}{642049175534} a^{18} + \frac{905952796609}{642049175534} a^{17} - \frac{1161390282979}{642049175534} a^{16} - \frac{357702170538}{321024587767} a^{15} + \frac{2498926876769}{642049175534} a^{14} - \frac{113191209427}{642049175534} a^{13} - \frac{2272040097961}{642049175534} a^{12} + \frac{6528721896841}{642049175534} a^{11} - \frac{3028967188697}{321024587767} a^{10} - \frac{1301307675829}{642049175534} a^{9} + \frac{2397131374696}{321024587767} a^{8} + \frac{4907832161917}{642049175534} a^{7} - \frac{3756538656239}{321024587767} a^{6} - \frac{1108670333727}{321024587767} a^{5} + \frac{232868198816}{321024587767} a^{4} + \frac{1595906775381}{321024587767} a^{3} + \frac{256355337011}{321024587767} a^{2} + \frac{198070685613}{321024587767} a - \frac{1044243418317}{642049175534} \),  \( \frac{28192748503}{642049175534} a^{19} - \frac{631616852431}{642049175534} a^{18} + \frac{941450397995}{321024587767} a^{17} - \frac{3150348279657}{642049175534} a^{16} + \frac{2001723654635}{321024587767} a^{15} + \frac{787859571471}{642049175534} a^{14} - \frac{6737652977769}{642049175534} a^{13} + \frac{3082910349399}{642049175534} a^{12} + \frac{2395089113581}{321024587767} a^{11} - \frac{19212003952043}{642049175534} a^{10} + \frac{10925813358836}{321024587767} a^{9} - \frac{2326239940862}{321024587767} a^{8} - \frac{5036054872925}{321024587767} a^{7} - \frac{3812931663790}{321024587767} a^{6} + \frac{19468727262271}{642049175534} a^{5} - \frac{100192757131}{642049175534} a^{4} - \frac{275762804047}{49388398118} a^{3} - \frac{3152083332993}{642049175534} a^{2} + \frac{10828715217}{24694199059} a + \frac{121618078147}{321024587767} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 77.3255035829 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-7}, \sqrt{17})\), 5.1.14161.1 x5, 10.0.23863536599.2, 10.0.1403737447.1 x5, 10.2.3409076657.1 x5

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

Sibling fields

Degree 10 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$