Properties

Label 20.0.25314910362...8272.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{10}\cdot 47^{11}$
Root discriminant $11.75$
Ramified primes $2, 47$
Class number $1$
Class group Trivial
Galois Group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 0, 3, 0, 4, -4, 32, -1, -34, 34, 9, -17, 0, 19, -17, 2, 7, -3, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 3*x^18 + 7*x^17 + 2*x^16 - 17*x^15 + 19*x^14 - 17*x^12 + 9*x^11 + 34*x^10 - 34*x^9 - x^8 + 32*x^7 - 4*x^6 + 4*x^5 + 3*x^3 - 3*x + 1)
gp: K = bnfinit(x^20 - x^19 - 3*x^18 + 7*x^17 + 2*x^16 - 17*x^15 + 19*x^14 - 17*x^12 + 9*x^11 + 34*x^10 - 34*x^9 - x^8 + 32*x^7 - 4*x^6 + 4*x^5 + 3*x^3 - 3*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut x^{19} \) \(\mathstrut -\mathstrut 3 x^{18} \) \(\mathstrut +\mathstrut 7 x^{17} \) \(\mathstrut +\mathstrut 2 x^{16} \) \(\mathstrut -\mathstrut 17 x^{15} \) \(\mathstrut +\mathstrut 19 x^{14} \) \(\mathstrut -\mathstrut 17 x^{12} \) \(\mathstrut +\mathstrut 9 x^{11} \) \(\mathstrut +\mathstrut 34 x^{10} \) \(\mathstrut -\mathstrut 34 x^{9} \) \(\mathstrut -\mathstrut x^{8} \) \(\mathstrut +\mathstrut 32 x^{7} \) \(\mathstrut -\mathstrut 4 x^{6} \) \(\mathstrut +\mathstrut 4 x^{5} \) \(\mathstrut +\mathstrut 3 x^{3} \) \(\mathstrut -\mathstrut 3 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(2531491036246028598272=2^{10}\cdot 47^{11}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $11.75$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 47$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} + \frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{14846564464} a^{19} + \frac{46617995}{3711641116} a^{18} - \frac{1360200467}{14846564464} a^{17} - \frac{113637093}{3711641116} a^{16} + \frac{320410967}{7423282232} a^{15} - \frac{48263067}{14846564464} a^{14} + \frac{751002127}{3711641116} a^{13} - \frac{142079382}{927910279} a^{12} + \frac{2250779955}{14846564464} a^{11} + \frac{1085169913}{3711641116} a^{10} + \frac{3394757099}{7423282232} a^{9} - \frac{1579194523}{3711641116} a^{8} - \frac{3604907821}{14846564464} a^{7} + \frac{2824830847}{14846564464} a^{6} - \frac{1360668653}{14846564464} a^{5} - \frac{2514755405}{14846564464} a^{4} + \frac{11159563}{14846564464} a^{3} + \frac{3402368337}{7423282232} a^{2} - \frac{1986269885}{7423282232} a - \frac{4886619201}{14846564464}$

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:  \( \frac{42930541143}{14846564464} a^{19} - \frac{4396494632}{927910279} a^{18} - \frac{79743356081}{14846564464} a^{17} + \frac{85823339067}{3711641116} a^{16} - \frac{70354303691}{7423282232} a^{15} - \frac{602158670765}{14846564464} a^{14} + \frac{147533343347}{1855820558} a^{13} - \frac{205184662115}{3711641116} a^{12} - \frac{72801703935}{14846564464} a^{11} + \frac{86131756387}{3711641116} a^{10} + \frac{608926949357}{7423282232} a^{9} - \frac{546418503325}{3711641116} a^{8} + \frac{1472227126821}{14846564464} a^{7} + \frac{172294714405}{14846564464} a^{6} - \frac{126930283723}{14846564464} a^{5} + \frac{295546862905}{14846564464} a^{4} - \frac{241518656603}{14846564464} a^{3} + \frac{142320421793}{7423282232} a^{2} - \frac{111768717217}{7423282232} a + \frac{39928956277}{14846564464} \),  \( \frac{7621670717}{14846564464} a^{19} - \frac{923536107}{7423282232} a^{18} - \frac{25629781985}{14846564464} a^{17} + \frac{2159674280}{927910279} a^{16} + \frac{22608841433}{7423282232} a^{15} - \frac{101738694095}{14846564464} a^{14} + \frac{31510628157}{7423282232} a^{13} + \frac{31176730759}{7423282232} a^{12} - \frac{97877483411}{14846564464} a^{11} - \frac{430464185}{3711641116} a^{10} + \frac{130243170659}{7423282232} a^{9} - \frac{3920002866}{927910279} a^{8} - \frac{85137180589}{14846564464} a^{7} + \frac{184907310969}{14846564464} a^{6} + \frac{103529526207}{14846564464} a^{5} + \frac{98108999209}{14846564464} a^{4} + \frac{57355294883}{14846564464} a^{3} + \frac{3350156297}{1855820558} a^{2} + \frac{1939136653}{927910279} a - \frac{8695128335}{14846564464} \),  \( \frac{19052758785}{14846564464} a^{19} - \frac{2652576985}{3711641116} a^{18} - \frac{63402790719}{14846564464} a^{17} + \frac{26835081229}{3711641116} a^{16} + \frac{45005632379}{7423282232} a^{15} - \frac{296734346435}{14846564464} a^{14} + \frac{57504618673}{3711641116} a^{13} + \frac{32746171513}{3711641116} a^{12} - \frac{302515832389}{14846564464} a^{11} + \frac{5290254025}{1855820558} a^{10} + \frac{347526262605}{7423282232} a^{9} - \frac{22548850475}{927910279} a^{8} - \frac{230424179633}{14846564464} a^{7} + \frac{577859480515}{14846564464} a^{6} + \frac{174563144003}{14846564464} a^{5} + \frac{91692445547}{14846564464} a^{4} + \frac{61969999999}{14846564464} a^{3} + \frac{37057535251}{7423282232} a^{2} + \frac{18970509643}{7423282232} a - \frac{37635093973}{14846564464} \),  \( \frac{135466377}{77730704} a^{19} - \frac{164557893}{38865352} a^{18} - \frac{95351069}{77730704} a^{17} + \frac{82189081}{4858169} a^{16} - \frac{633137551}{38865352} a^{15} - \frac{1710323595}{77730704} a^{14} + \frac{2652230079}{38865352} a^{13} - \frac{2625910709}{38865352} a^{12} + \frac{1235245437}{77730704} a^{11} + \frac{103503498}{4858169} a^{10} + \frac{1544582013}{38865352} a^{9} - \frac{2549196653}{19432676} a^{8} + \frac{9576050883}{77730704} a^{7} - \frac{2044097171}{77730704} a^{6} - \frac{1588539877}{77730704} a^{5} + \frac{1049955593}{77730704} a^{4} - \frac{1233056661}{77730704} a^{3} + \frac{169751839}{9716338} a^{2} - \frac{340858653}{19432676} a + \frac{511271709}{77730704} \),  \( a \),  \( \frac{19810874867}{7423282232} a^{19} - \frac{32594775139}{7423282232} a^{18} - \frac{8866633441}{1855820558} a^{17} + \frac{78598235091}{3711641116} a^{16} - \frac{8558304155}{927910279} a^{15} - \frac{268784939021}{7423282232} a^{14} + \frac{545010084809}{7423282232} a^{13} - \frac{399646367311}{7423282232} a^{12} - \frac{1311374457}{3711641116} a^{11} + \frac{36812974319}{1855820558} a^{10} + \frac{274608487997}{3711641116} a^{9} - \frac{499382422995}{3711641116} a^{8} + \frac{719312539135}{7423282232} a^{7} + \frac{13336300315}{3711641116} a^{6} - \frac{36637046591}{7423282232} a^{5} + \frac{41159006395}{1855820558} a^{4} - \frac{118104831157}{7423282232} a^{3} + \frac{144908537347}{7423282232} a^{2} - \frac{101696674515}{7423282232} a + \frac{14505635571}{3711641116} \),  \( \frac{19467789525}{14846564464} a^{19} - \frac{10880815309}{3711641116} a^{18} - \frac{18937745687}{14846564464} a^{17} + \frac{45065678199}{3711641116} a^{16} - \frac{76271363169}{7423282232} a^{15} - \frac{250455922031}{14846564464} a^{14} + \frac{175569143775}{3711641116} a^{13} - \frac{82540598527}{1855820558} a^{12} + \frac{138996648799}{14846564464} a^{11} + \frac{49425748015}{3711641116} a^{10} + \frac{239894085555}{7423282232} a^{9} - \frac{334779468925}{3711641116} a^{8} + \frac{1199075570543}{14846564464} a^{7} - \frac{223526267909}{14846564464} a^{6} - \frac{140747332025}{14846564464} a^{5} + \frac{127582572519}{14846564464} a^{4} - \frac{166693459825}{14846564464} a^{3} + \frac{101802582609}{7423282232} a^{2} - \frac{85266870981}{7423282232} a + \frac{61603598475}{14846564464} \),  \( \frac{27248764839}{7423282232} a^{19} - \frac{47683015085}{7423282232} a^{18} - \frac{22680995327}{3711641116} a^{17} + \frac{112549157363}{3711641116} a^{16} - \frac{28960692713}{1855820558} a^{15} - \frac{375661809113}{7423282232} a^{14} + \frac{802900734219}{7423282232} a^{13} - \frac{603958558401}{7423282232} a^{12} - \frac{1004796874}{927910279} a^{11} + \frac{31324412425}{927910279} a^{10} + \frac{370151920369}{3711641116} a^{9} - \frac{742728463295}{3711641116} a^{8} + \frac{1103308884843}{7423282232} a^{7} + \frac{6568703338}{927910279} a^{6} - \frac{136052493675}{7423282232} a^{5} + \frac{106596516153}{3711641116} a^{4} - \frac{150792598301}{7423282232} a^{3} + \frac{209496088269}{7423282232} a^{2} - \frac{137581850465}{7423282232} a + \frac{7978300661}{1855820558} \),  \( \frac{13303345255}{14846564464} a^{19} - \frac{5786119197}{1855820558} a^{18} + \frac{11126898307}{14846564464} a^{17} + \frac{39814666267}{3711641116} a^{16} - \frac{116014441431}{7423282232} a^{15} - \frac{141671096853}{14846564464} a^{14} + \frac{45405349674}{927910279} a^{13} - \frac{108487056967}{1855820558} a^{12} + \frac{320283175241}{14846564464} a^{11} + \frac{30014409461}{1855820558} a^{10} + \frac{93069686391}{7423282232} a^{9} - \frac{176823340059}{1855820558} a^{8} + \frac{1588610478913}{14846564464} a^{7} - \frac{557204199951}{14846564464} a^{6} - \frac{302505293355}{14846564464} a^{5} + \frac{145280398905}{14846564464} a^{4} - \frac{212511778647}{14846564464} a^{3} + \frac{93752354183}{7423282232} a^{2} - \frac{105891932353}{7423282232} a + \frac{89259206161}{14846564464} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 179.782999261 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

20T423:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{-47}) \), 5.1.2209.1 x5, 10.0.229345007.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 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 R ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.10.10$x^{10} - 11 x^{8} + 10 x^{6} - 62 x^{4} + 21 x^{2} - 55$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
$47$47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.3.1$x^{4} + 94$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$
47.4.2.1$x^{4} + 1175 x^{2} + 373321$$2$$2$$2$$V_4$$[\ ]_{2}^{2}$