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Properties

Label 20.20.1768769550...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{26}\cdot 61^{8}$
Root discriminant $145.34$
Ramified primes $2, 3, 5, 61$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5^2$ (as 20T28)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![916095316, 1007795640, -8806862800, -681612780, 10914530225, 784838688, -4729220200, -221247000, 1009834870, 25437000, -122060112, -1405800, 8908075, 37080, -400000, -372, 10790, 0, -160, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316)
 
gp: K = bnfinit(x^20 - 160*x^18 + 10790*x^16 - 372*x^15 - 400000*x^14 + 37080*x^13 + 8908075*x^12 - 1405800*x^11 - 122060112*x^10 + 25437000*x^9 + 1009834870*x^8 - 221247000*x^7 - 4729220200*x^6 + 784838688*x^5 + 10914530225*x^4 - 681612780*x^3 - 8806862800*x^2 + 1007795640*x + 916095316, 1)
 

Normalized defining polynomial

\( x^{20} - 160 x^{18} + 10790 x^{16} - 372 x^{15} - 400000 x^{14} + 37080 x^{13} + 8908075 x^{12} - 1405800 x^{11} - 122060112 x^{10} + 25437000 x^{9} + 1009834870 x^{8} - 221247000 x^{7} - 4729220200 x^{6} + 784838688 x^{5} + 10914530225 x^{4} - 681612780 x^{3} - 8806862800 x^{2} + 1007795640 x + 916095316 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17687695508088196514062500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{26}\cdot 61^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $145.34$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{10} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{5} + \frac{2}{5} a^{4} - \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3} + \frac{1}{10} a^{2} + \frac{2}{5}$, $\frac{1}{20} a^{10} - \frac{1}{4} a^{6} - \frac{3}{10} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{10}$, $\frac{1}{20} a^{11} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} + \frac{1}{4} a^{3} + \frac{3}{10} a$, $\frac{1}{60} a^{12} - \frac{1}{60} a^{10} - \frac{1}{20} a^{8} + \frac{1}{10} a^{7} - \frac{1}{12} a^{6} + \frac{2}{5} a^{5} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3} - \frac{19}{60} a^{2} - \frac{3}{10} a - \frac{7}{30}$, $\frac{1}{60} a^{13} - \frac{1}{60} a^{11} - \frac{1}{20} a^{9} - \frac{11}{60} a^{7} - \frac{1}{10} a^{6} + \frac{3}{20} a^{5} - \frac{1}{10} a^{4} - \frac{13}{60} a^{3} - \frac{3}{10} a^{2} + \frac{1}{6} a + \frac{2}{5}$, $\frac{1}{600} a^{14} - \frac{1}{300} a^{13} + \frac{1}{200} a^{12} + \frac{1}{75} a^{11} - \frac{1}{150} a^{10} + \frac{1}{25} a^{9} - \frac{23}{600} a^{8} - \frac{1}{75} a^{7} - \frac{17}{150} a^{6} + \frac{1}{25} a^{5} - \frac{139}{600} a^{4} - \frac{29}{75} a^{3} - \frac{19}{200} a^{2} + \frac{89}{300} a - \frac{127}{300}$, $\frac{1}{36600} a^{15} - \frac{221}{36600} a^{13} + \frac{1}{150} a^{12} - \frac{17}{9150} a^{11} + \frac{29}{9150} a^{10} - \frac{151}{7320} a^{9} - \frac{41}{1525} a^{8} + \frac{877}{4575} a^{7} + \frac{337}{9150} a^{6} + \frac{10709}{36600} a^{5} + \frac{2}{5} a^{4} + \frac{239}{600} a^{3} - \frac{19}{150} a^{2} - \frac{149}{300} a + \frac{13}{150}$, $\frac{1}{36600} a^{16} + \frac{23}{36600} a^{14} - \frac{1}{150} a^{13} + \frac{9}{6100} a^{12} + \frac{119}{18300} a^{11} + \frac{709}{36600} a^{10} + \frac{101}{3050} a^{9} - \frac{71}{6100} a^{8} + \frac{2443}{18300} a^{7} + \frac{2657}{36600} a^{6} + \frac{23}{50} a^{5} + \frac{13}{600} a^{4} + \frac{83}{300} a^{3} - \frac{41}{100} a^{2} + \frac{13}{75} a - \frac{4}{25}$, $\frac{1}{36600} a^{17} - \frac{77}{12200} a^{13} + \frac{119}{18300} a^{12} - \frac{3}{2440} a^{11} + \frac{1}{4575} a^{10} + \frac{167}{7320} a^{9} - \frac{9}{6100} a^{8} - \frac{891}{12200} a^{7} - \frac{379}{1830} a^{6} + \frac{2757}{6100} a^{5} + \frac{1}{20} a^{4} - \frac{7}{200} a^{3} + \frac{11}{150} a^{2} + \frac{1}{60} a + \frac{37}{150}$, $\frac{1}{174142800} a^{18} - \frac{17}{2854800} a^{17} + \frac{83}{6697800} a^{16} - \frac{19}{2854800} a^{15} - \frac{73573}{174142800} a^{14} - \frac{496543}{87071400} a^{13} + \frac{189077}{174142800} a^{12} - \frac{135631}{19349200} a^{11} - \frac{160701}{19349200} a^{10} + \frac{1024313}{29023800} a^{9} + \frac{2877727}{58047600} a^{8} - \frac{206029}{951600} a^{7} - \frac{21571}{109800} a^{6} + \frac{325699}{2854800} a^{5} - \frac{895913}{2854800} a^{4} - \frac{40097}{142740} a^{3} + \frac{877}{2340} a^{2} + \frac{5461}{11700} a - \frac{3727}{11700}$, $\frac{1}{2484741735498845196509900214378512375547771600} a^{19} - \frac{1265378932764658483546109067160704167}{828247245166281732169966738126170791849257200} a^{18} + \frac{5655458673884325300185245949973503294551}{414123622583140866084983369063085395924628600} a^{17} - \frac{3883524927554419159496063701489709050751}{2484741735498845196509900214378512375547771600} a^{16} + \frac{1619521534304330565699275222281026618887}{165649449033256346433993347625234158369851440} a^{15} + \frac{30862343593324309672182784753297707415043}{248474173549884519650990021437851237554777160} a^{14} + \frac{196958396422311131961171785566024099591}{667761820881173124565950071050393006059600} a^{13} - \frac{15083676366354243465086240507636334765371747}{2484741735498845196509900214378512375547771600} a^{12} - \frac{1516044289595713450547406751379693463534059}{828247245166281732169966738126170791849257200} a^{11} - \frac{69509344813405431870729029756066889576391}{5521648301108544881133111587507805278995048} a^{10} - \frac{33252572648222415033215148128886302903773}{4247421770083496062410085836544465599226960} a^{9} + \frac{10964550462033643304890745059722616569582481}{276082415055427244056655579375390263949752400} a^{8} - \frac{2925077843300277934576026045793375363197469}{20366735536875780299261477167036986684817800} a^{7} + \frac{333883426338524045184721570775472795199143}{4525941230416840066502550481563774818848400} a^{6} + \frac{269538077919564505115891232585422692290573}{905188246083368013300510096312754963769680} a^{5} - \frac{48358012260709162716525412320320072162143}{5091683884218945074815369291759246671204450} a^{4} + \frac{509508530225969124435052771255582167226867}{1357782369125052019950765144469132445654520} a^{3} - \frac{39786685682790630683042727026166103304179}{166940455220293281141487517762598251514900} a^{2} + \frac{20290215996068680213640653021958384992567}{41735113805073320285371879440649562878725} a - \frac{7054236352178434428126287936592210207281}{16694045522029328114148751776259825151490}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9814449721720000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5^2$ (as 20T28):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 16 conjugacy class representatives for $D_5^2$
Character table for $D_5^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}, \sqrt{5})\), 10.10.841134840750000000000.1 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
Degree 20 sibling: data not computed
Degree 25 sibling: 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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5Data not computed
61Data not computed