↑

Properties

Label 20.20.4527516723...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{16}\cdot 3^{16}\cdot 5^{18}\cdot 29^{10}$
Root discriminant $96.12$
Ramified primes $2, 3, 5, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_5\wr C_2$ (as 20T50)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7604, 66168, -147832, -1037404, -246308, 2635222, 801892, -3049052, -439446, 1820396, -58627, -533244, 68447, 81952, -14858, -6704, 1426, 270, -63, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 63*x^18 + 270*x^17 + 1426*x^16 - 6704*x^15 - 14858*x^14 + 81952*x^13 + 68447*x^12 - 533244*x^11 - 58627*x^10 + 1820396*x^9 - 439446*x^8 - 3049052*x^7 + 801892*x^6 + 2635222*x^5 - 246308*x^4 - 1037404*x^3 - 147832*x^2 + 66168*x + 7604)
 
gp: K = bnfinit(x^20 - 4*x^19 - 63*x^18 + 270*x^17 + 1426*x^16 - 6704*x^15 - 14858*x^14 + 81952*x^13 + 68447*x^12 - 533244*x^11 - 58627*x^10 + 1820396*x^9 - 439446*x^8 - 3049052*x^7 + 801892*x^6 + 2635222*x^5 - 246308*x^4 - 1037404*x^3 - 147832*x^2 + 66168*x + 7604, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 63 x^{18} + 270 x^{17} + 1426 x^{16} - 6704 x^{15} - 14858 x^{14} + 81952 x^{13} + 68447 x^{12} - 533244 x^{11} - 58627 x^{10} + 1820396 x^{9} - 439446 x^{8} - 3049052 x^{7} + 801892 x^{6} + 2635222 x^{5} - 246308 x^{4} - 1037404 x^{3} - 147832 x^{2} + 66168 x + 7604 \)

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:  \(4527516723638915422730250000000000000000=2^{16}\cdot 3^{16}\cdot 5^{18}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.12$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{10} a^{13} + \frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{2}{5} a^{6} + \frac{3}{10} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{10} a^{14} + \frac{1}{10} a^{12} + \frac{1}{5} a^{11} - \frac{1}{10} a^{10} - \frac{1}{5} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{150} a^{15} + \frac{1}{25} a^{14} + \frac{1}{75} a^{13} + \frac{1}{6} a^{12} + \frac{1}{150} a^{11} - \frac{11}{150} a^{10} + \frac{7}{30} a^{9} + \frac{1}{50} a^{8} + \frac{16}{75} a^{7} + \frac{23}{50} a^{6} - \frac{1}{30} a^{5} - \frac{32}{75} a^{4} + \frac{3}{25} a^{3} + \frac{7}{75} a^{2} + \frac{37}{75} a - \frac{29}{75}$, $\frac{1}{150} a^{16} - \frac{2}{75} a^{14} - \frac{1}{75} a^{13} + \frac{1}{150} a^{12} - \frac{16}{75} a^{11} - \frac{17}{75} a^{10} + \frac{21}{50} a^{9} + \frac{37}{75} a^{8} + \frac{12}{25} a^{7} + \frac{23}{75} a^{6} + \frac{71}{150} a^{5} + \frac{12}{25} a^{4} - \frac{32}{75} a^{3} - \frac{7}{15} a^{2} + \frac{19}{75} a + \frac{8}{25}$, $\frac{1}{1050} a^{17} + \frac{1}{350} a^{16} + \frac{1}{1050} a^{15} + \frac{23}{525} a^{14} - \frac{1}{105} a^{13} + \frac{57}{350} a^{12} - \frac{43}{210} a^{11} + \frac{71}{1050} a^{10} - \frac{57}{175} a^{9} + \frac{24}{175} a^{8} - \frac{254}{525} a^{7} - \frac{331}{1050} a^{6} + \frac{43}{210} a^{5} + \frac{17}{175} a^{4} + \frac{79}{525} a^{3} + \frac{23}{175} a^{2} + \frac{23}{75} a + \frac{152}{525}$, $\frac{1}{54367950} a^{18} + \frac{6481}{54367950} a^{17} + \frac{42782}{27183975} a^{16} + \frac{8441}{2588950} a^{15} + \frac{35283}{18122650} a^{14} + \frac{107006}{27183975} a^{13} - \frac{2985223}{54367950} a^{12} - \frac{311401}{1553370} a^{11} - \frac{1102459}{18122650} a^{10} - \frac{133663}{310674} a^{9} - \frac{25962439}{54367950} a^{8} + \frac{13406}{1294475} a^{7} + \frac{22037329}{54367950} a^{6} - \frac{4407394}{9061325} a^{5} + \frac{11075072}{27183975} a^{4} + \frac{9629506}{27183975} a^{3} - \frac{861526}{9061325} a^{2} + \frac{839518}{5436795} a + \frac{3443718}{9061325}$, $\frac{1}{91987780090053748126494450} a^{19} - \frac{745916708105752121}{91987780090053748126494450} a^{18} + \frac{1407946293789423602631}{15331296681675624687749075} a^{17} - \frac{133772798174627975294}{505427363132163451244475} a^{16} + \frac{126382372400658278271271}{91987780090053748126494450} a^{15} - \frac{140615261477172284585837}{3679511203602149925059778} a^{14} - \frac{1107593237309198513665667}{30662593363351249375498150} a^{13} - \frac{1278308967019800590446337}{6570555720718124866178175} a^{12} - \frac{7263754190178192059447417}{30662593363351249375498150} a^{11} + \frac{16277934447946620589059}{438037048047874991078545} a^{10} + \frac{11800246817419812328121053}{30662593363351249375498150} a^{9} + \frac{237225729855011946212648}{938650817245446409454025} a^{8} + \frac{11308592375839377503102479}{30662593363351249375498150} a^{7} - \frac{1570616104184115243449899}{15331296681675624687749075} a^{6} - \frac{14185478067000420832727753}{45993890045026874063247225} a^{5} + \frac{6525474955859563586226236}{15331296681675624687749075} a^{4} + \frac{1534572752451600284436289}{3537991541925144158711325} a^{3} + \frac{560401976783789579667659}{15331296681675624687749075} a^{2} - \frac{12412757765169789662341489}{45993890045026874063247225} a + \frac{84664511691096174290279}{312883605748482136484675}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 42038324782300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\), 10.10.538294594356000000.1

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 siblings: data not computed
Degree 25 sibling: 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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ R ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$