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Properties

Label 20.0.31241493709...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 17^{10}$
Root discriminant $59.53$
Ramified primes $2, 3, 5, 17$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4345624, 6029496, -550760, -1873344, 4162668, 2515542, -2510994, -1408896, 914778, 344478, -269871, -39048, 60849, 324, -8634, 204, 780, -12, -41, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 41*x^18 - 12*x^17 + 780*x^16 + 204*x^15 - 8634*x^14 + 324*x^13 + 60849*x^12 - 39048*x^11 - 269871*x^10 + 344478*x^9 + 914778*x^8 - 1408896*x^7 - 2510994*x^6 + 2515542*x^5 + 4162668*x^4 - 1873344*x^3 - 550760*x^2 + 6029496*x + 4345624)
 
gp: K = bnfinit(x^20 - 41*x^18 - 12*x^17 + 780*x^16 + 204*x^15 - 8634*x^14 + 324*x^13 + 60849*x^12 - 39048*x^11 - 269871*x^10 + 344478*x^9 + 914778*x^8 - 1408896*x^7 - 2510994*x^6 + 2515542*x^5 + 4162668*x^4 - 1873344*x^3 - 550760*x^2 + 6029496*x + 4345624, 1)
 

Normalized defining polynomial

\( x^{20} - 41 x^{18} - 12 x^{17} + 780 x^{16} + 204 x^{15} - 8634 x^{14} + 324 x^{13} + 60849 x^{12} - 39048 x^{11} - 269871 x^{10} + 344478 x^{9} + 914778 x^{8} - 1408896 x^{7} - 2510994 x^{6} + 2515542 x^{5} + 4162668 x^{4} - 1873344 x^{3} - 550760 x^{2} + 6029496 x + 4345624 \)

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:  \(312414937093187559824400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.53$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{12} - \frac{1}{6} a^{9} - \frac{1}{18} a^{8} + \frac{1}{3} a^{7} - \frac{1}{18} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{13} + \frac{1}{9} a^{9} - \frac{7}{18} a^{7} - \frac{1}{2} a^{5} - \frac{4}{9} a^{3} - \frac{4}{9} a$, $\frac{1}{180} a^{16} + \frac{1}{45} a^{15} + \frac{1}{18} a^{13} + \frac{7}{90} a^{12} - \frac{1}{30} a^{11} + \frac{1}{90} a^{10} + \frac{1}{9} a^{9} + \frac{7}{60} a^{8} + \frac{37}{90} a^{7} - \frac{17}{45} a^{6} + \frac{4}{15} a^{5} - \frac{13}{90} a^{4} - \frac{1}{90} a^{3} - \frac{2}{15} a^{2} - \frac{2}{45} a - \frac{16}{45}$, $\frac{1}{180} a^{17} + \frac{1}{45} a^{15} - \frac{1}{30} a^{13} - \frac{1}{15} a^{12} - \frac{1}{45} a^{11} + \frac{1}{15} a^{10} - \frac{19}{180} a^{9} + \frac{11}{30} a^{7} - \frac{1}{2} a^{6} + \frac{11}{90} a^{5} - \frac{13}{30} a^{4} + \frac{1}{45} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{15}$, $\frac{1}{180} a^{18} + \frac{1}{45} a^{15} + \frac{1}{45} a^{14} - \frac{1}{90} a^{13} + \frac{1}{18} a^{12} + \frac{1}{30} a^{11} + \frac{1}{60} a^{10} - \frac{1}{18} a^{9} - \frac{7}{45} a^{8} + \frac{11}{45} a^{7} + \frac{7}{90} a^{6} - \frac{1}{6} a^{5} + \frac{4}{15} a^{4} + \frac{19}{45} a^{3} + \frac{1}{45} a^{2} + \frac{7}{45} a - \frac{1}{45}$, $\frac{1}{23567110071666239455051501108650650251684535275307940} a^{19} + \frac{3850216907557366078057587945651342924635590195339}{7855703357222079818350500369550216750561511758435980} a^{18} + \frac{8082658473162009251798696260571512652446021148897}{11783555035833119727525750554325325125842267637653970} a^{17} - \frac{14617137605614709390561826907325431846012663392097}{5891777517916559863762875277162662562921133818826985} a^{16} + \frac{9090476338325695007193288238755412123760324346785}{785570335722207981835050036955021675056151175843598} a^{15} - \frac{16373289007901636391902382212114854764424900266047}{1963925839305519954587625092387554187640377939608995} a^{14} - \frac{296903334445585309111455810870020106155954126242417}{5891777517916559863762875277162662562921133818826985} a^{13} + \frac{64386790158417304517720057768444300841571795196807}{1178355503583311972752575055432532512584226763765397} a^{12} + \frac{725577443545493059022533862635442378378027860247111}{23567110071666239455051501108650650251684535275307940} a^{11} - \frac{185557325656631283408624989101514450374853846703187}{4713422014333247891010300221730130050336907055061588} a^{10} - \frac{61527664083852327776150070239471963159237629108173}{1309283892870346636391750061591702791760251959739330} a^{9} - \frac{295104576272932619958561009532067734065688606859182}{1963925839305519954587625092387554187640377939608995} a^{8} - \frac{1504107873731857392192742534311233856061012745848951}{11783555035833119727525750554325325125842267637653970} a^{7} + \frac{4991845916899621526008453575729835138159254838410817}{11783555035833119727525750554325325125842267637653970} a^{6} + \frac{18230709549892599254884633748200842027348354470549}{1178355503583311972752575055432532512584226763765397} a^{5} - \frac{625671283825396861380027198746100108491648035379317}{5891777517916559863762875277162662562921133818826985} a^{4} + \frac{887111565438623426545665514359422621171205180186583}{1963925839305519954587625092387554187640377939608995} a^{3} - \frac{746829815155767865686055609807218598981976196941882}{1963925839305519954587625092387554187640377939608995} a^{2} - \frac{66389561753023752381362792612968018442247481073585}{392785167861103990917525018477510837528075587921799} a - \frac{2100783741516440511950917408392399731610035592026294}{5891777517916559863762875277162662562921133818826985}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 4049681427.5162997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}, \sqrt{17})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.558940906620000000.1, 10.2.37262727108000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
3Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$17$17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$