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Properties

Label 20.0.17530598928...2384.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{26}\cdot 13^{7}\cdot 457^{7}$
Root discriminant $51.55$
Ramified primes $2, 13, 457$
Class number $300$ (GRH)
Class group $[2, 150]$ (GRH)
Galois group $C_2\times S_5$ (as 20T65)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36524, -49248, 21840, -49864, 54648, -15248, 12100, -19948, 7427, 24, 3668, -740, -157, -208, 204, -36, 53, -24, 12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 12*x^18 - 24*x^17 + 53*x^16 - 36*x^15 + 204*x^14 - 208*x^13 - 157*x^12 - 740*x^11 + 3668*x^10 + 24*x^9 + 7427*x^8 - 19948*x^7 + 12100*x^6 - 15248*x^5 + 54648*x^4 - 49864*x^3 + 21840*x^2 - 49248*x + 36524)
 
gp: K = bnfinit(x^20 - 4*x^19 + 12*x^18 - 24*x^17 + 53*x^16 - 36*x^15 + 204*x^14 - 208*x^13 - 157*x^12 - 740*x^11 + 3668*x^10 + 24*x^9 + 7427*x^8 - 19948*x^7 + 12100*x^6 - 15248*x^5 + 54648*x^4 - 49864*x^3 + 21840*x^2 - 49248*x + 36524, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 12 x^{18} - 24 x^{17} + 53 x^{16} - 36 x^{15} + 204 x^{14} - 208 x^{13} - 157 x^{12} - 740 x^{11} + 3668 x^{10} + 24 x^{9} + 7427 x^{8} - 19948 x^{7} + 12100 x^{6} - 15248 x^{5} + 54648 x^{4} - 49864 x^{3} + 21840 x^{2} - 49248 x + 36524 \)

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:  \(17530598928152643195807706208272384=2^{26}\cdot 13^{7}\cdot 457^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.55$
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:  $2, 13, 457$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{7} + \frac{3}{16} a^{6} - \frac{1}{16} a^{5} + \frac{7}{16} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{9} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{16} a^{14} - \frac{3}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{15} - \frac{3}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{16} - \frac{1}{8} a^{9} + \frac{1}{32} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{32} a^{17} + \frac{1}{32} a^{9} - \frac{1}{4} a^{7} - \frac{5}{16} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{128} a^{18} - \frac{1}{64} a^{17} - \frac{1}{128} a^{16} - \frac{1}{64} a^{14} - \frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{3}{128} a^{10} - \frac{3}{64} a^{9} + \frac{3}{128} a^{8} + \frac{7}{32} a^{7} - \frac{1}{4} a^{6} - \frac{15}{32} a^{4} - \frac{1}{16} a^{3} - \frac{11}{32} a^{2} + \frac{3}{16} a - \frac{5}{32}$, $\frac{1}{46281289221806690135477095400813312} a^{19} - \frac{145784227381765998208234787319533}{46281289221806690135477095400813312} a^{18} - \frac{137870400786295176011894952748635}{46281289221806690135477095400813312} a^{17} - \frac{645619678091309775496402550974365}{46281289221806690135477095400813312} a^{16} - \frac{404186627871478238354627418150309}{23140644610903345067738547700406656} a^{15} - \frac{552194940704921957577898767880349}{23140644610903345067738547700406656} a^{14} + \frac{708212394718749349887076226116607}{23140644610903345067738547700406656} a^{13} - \frac{50003151799169444307204979205439}{23140644610903345067738547700406656} a^{12} + \frac{687752133357336609706606135391305}{46281289221806690135477095400813312} a^{11} - \frac{1455601600838344999361782987937477}{46281289221806690135477095400813312} a^{10} + \frac{2917263462162764818120762855156949}{46281289221806690135477095400813312} a^{9} + \frac{5761342717225016041942323950206899}{46281289221806690135477095400813312} a^{8} - \frac{1181016715310813055500511031885207}{11570322305451672533869273850203328} a^{7} + \frac{408183737089170434484605184253087}{2892580576362918133467318462550832} a^{6} + \frac{1184345311100455341566744959980857}{11570322305451672533869273850203328} a^{5} + \frac{1629331118482266848801907448931115}{11570322305451672533869273850203328} a^{4} - \frac{2578992139931651573449467246636841}{11570322305451672533869273850203328} a^{3} + \frac{1083176694513515469444071954335743}{11570322305451672533869273850203328} a^{2} - \frac{1931905892553522310849373461761687}{11570322305451672533869273850203328} a + \frac{697917364633500826374165978224623}{11570322305451672533869273850203328}$

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

Class group and class number

$C_{2}\times C_{150}$, which has order $300$ (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:  $9$
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:  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:  \( 27454988.0022 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_5$ (as 20T65):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 240
The 14 conjugacy class representatives for $C_2\times S_5$
Character table for $C_2\times S_5$

Intermediate fields

10.10.858892093935616.1

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 12 siblings: data not computed
Degree 20 siblings: data not computed
Degree 24 siblings: data not computed
Degree 30 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
457Data not computed