↑

Properties

Label 20.4.42441592211...0625.2
Degree $20$
Signature $[4, 8]$
Discriminant $3^{14}\cdot 5^{14}\cdot 7^{14}\cdot 11^{8}$
Root discriminant $67.82$
Ramified primes $3, 5, 7, 11$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $C_5^2:Q_8$ (as 20T47)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![103525, 314550, 219900, -236625, -745940, -370575, 1068910, 711180, -366539, -379482, -90176, 54018, 40177, 9342, 1054, -1221, -236, -99, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 12*x^18 - 99*x^17 - 236*x^16 - 1221*x^15 + 1054*x^14 + 9342*x^13 + 40177*x^12 + 54018*x^11 - 90176*x^10 - 379482*x^9 - 366539*x^8 + 711180*x^7 + 1068910*x^6 - 370575*x^5 - 745940*x^4 - 236625*x^3 + 219900*x^2 + 314550*x + 103525)
 
gp: K = bnfinit(x^20 + 12*x^18 - 99*x^17 - 236*x^16 - 1221*x^15 + 1054*x^14 + 9342*x^13 + 40177*x^12 + 54018*x^11 - 90176*x^10 - 379482*x^9 - 366539*x^8 + 711180*x^7 + 1068910*x^6 - 370575*x^5 - 745940*x^4 - 236625*x^3 + 219900*x^2 + 314550*x + 103525, 1)
 

Normalized defining polynomial

\( x^{20} + 12 x^{18} - 99 x^{17} - 236 x^{16} - 1221 x^{15} + 1054 x^{14} + 9342 x^{13} + 40177 x^{12} + 54018 x^{11} - 90176 x^{10} - 379482 x^{9} - 366539 x^{8} + 711180 x^{7} + 1068910 x^{6} - 370575 x^{5} - 745940 x^{4} - 236625 x^{3} + 219900 x^{2} + 314550 x + 103525 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4244159221123694514715246343994140625=3^{14}\cdot 5^{14}\cdot 7^{14}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.82$
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:  $3, 5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{6}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a$, $\frac{1}{60} a^{10} - \frac{1}{12} a^{9} + \frac{1}{60} a^{8} + \frac{1}{20} a^{7} - \frac{11}{60} a^{6} - \frac{13}{30} a^{5} + \frac{1}{60} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{12} a - \frac{1}{12}$, $\frac{1}{60} a^{11} - \frac{1}{15} a^{9} - \frac{1}{30} a^{8} + \frac{1}{15} a^{7} - \frac{7}{20} a^{6} - \frac{19}{60} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a + \frac{5}{12}$, $\frac{1}{180} a^{12} + \frac{1}{180} a^{10} + \frac{1}{60} a^{9} - \frac{1}{180} a^{8} - \frac{1}{30} a^{7} - \frac{11}{45} a^{6} - \frac{1}{6} a^{5} - \frac{17}{36} a^{4} + \frac{5}{12} a^{3} - \frac{13}{36} a^{2} + \frac{1}{6} a + \frac{17}{36}$, $\frac{1}{360} a^{13} - \frac{1}{360} a^{12} + \frac{1}{360} a^{11} + \frac{1}{180} a^{10} + \frac{13}{180} a^{9} + \frac{5}{72} a^{8} - \frac{19}{180} a^{7} + \frac{13}{45} a^{6} + \frac{31}{72} a^{5} + \frac{1}{36} a^{4} - \frac{5}{36} a^{3} - \frac{17}{72} a^{2} - \frac{19}{72} a - \frac{11}{72}$, $\frac{1}{1080} a^{14} + \frac{1}{360} a^{11} - \frac{1}{540} a^{10} + \frac{3}{40} a^{9} - \frac{43}{1080} a^{8} - \frac{17}{90} a^{7} + \frac{409}{1080} a^{6} - \frac{5}{72} a^{5} - \frac{7}{108} a^{4} + \frac{7}{24} a^{3} - \frac{13}{36} a^{2} + \frac{7}{18} a - \frac{89}{216}$, $\frac{1}{2160} a^{15} - \frac{1}{720} a^{13} - \frac{1}{360} a^{12} + \frac{13}{2160} a^{11} + \frac{1}{240} a^{10} - \frac{139}{2160} a^{9} + \frac{1}{720} a^{8} + \frac{101}{432} a^{7} + \frac{299}{720} a^{6} + \frac{167}{2160} a^{5} - \frac{101}{240} a^{4} + \frac{13}{72} a^{3} + \frac{7}{144} a^{2} - \frac{61}{216} a - \frac{19}{48}$, $\frac{1}{30240} a^{16} - \frac{1}{6048} a^{15} - \frac{1}{10080} a^{14} + \frac{1}{2016} a^{13} - \frac{47}{30240} a^{12} + \frac{13}{3024} a^{11} - \frac{11}{1512} a^{10} - \frac{149}{15120} a^{9} - \frac{31}{378} a^{8} + \frac{1759}{7560} a^{7} + \frac{3853}{15120} a^{6} - \frac{1021}{3024} a^{5} - \frac{4943}{10080} a^{4} - \frac{17}{288} a^{3} - \frac{1865}{6048} a^{2} - \frac{113}{864} a + \frac{107}{224}$, $\frac{1}{30240} a^{17} + \frac{1}{1080} a^{13} - \frac{1}{1440} a^{12} - \frac{13}{3024} a^{11} - \frac{23}{5040} a^{10} + \frac{941}{15120} a^{9} - \frac{83}{2520} a^{8} + \frac{1969}{15120} a^{7} + \frac{211}{2520} a^{6} + \frac{887}{2016} a^{5} - \frac{433}{5040} a^{4} - \frac{397}{3024} a^{3} + \frac{193}{504} a^{2} + \frac{247}{1008} a - \frac{19}{672}$, $\frac{1}{453600} a^{18} + \frac{1}{151200} a^{17} + \frac{1}{453600} a^{16} + \frac{1}{50400} a^{15} - \frac{199}{453600} a^{14} - \frac{1}{9450} a^{13} - \frac{1}{420} a^{12} - \frac{13}{12600} a^{11} - \frac{1861}{226800} a^{10} - \frac{5003}{75600} a^{9} + \frac{1327}{22680} a^{8} - \frac{5441}{37800} a^{7} + \frac{14243}{30240} a^{6} + \frac{5071}{30240} a^{5} - \frac{2677}{12960} a^{4} - \frac{13609}{30240} a^{3} + \frac{8243}{18144} a^{2} + \frac{25}{56} a - \frac{251}{9072}$, $\frac{1}{15820677094321114413270621375513600} a^{19} - \frac{656406648479568892580800243}{753365575920053067298601017881600} a^{18} - \frac{173846783790698459080538173907}{15820677094321114413270621375513600} a^{17} - \frac{3270163748897391759788223419}{878926505240061911848367854195200} a^{16} - \frac{1408329419453723100783213260729}{7910338547160557206635310687756800} a^{15} + \frac{295330559104985832976102592929}{1757853010480123823696735708390400} a^{14} + \frac{1132823656186000499520546634781}{5273559031440371471090207125171200} a^{13} + \frac{5805506928164090867139267530519}{5273559031440371471090207125171200} a^{12} + \frac{41667567528536369736420135735203}{7910338547160557206635310687756800} a^{11} - \frac{3591627049214106789410880061}{3487803592222467904160189897600} a^{10} + \frac{15271973294088818664305450952283}{282512090970019900236975381705600} a^{9} + \frac{7473720696148456475556877881797}{292975501746687303949455951398400} a^{8} + \frac{31036421892681040720418996926283}{1757853010480123823696735708390400} a^{7} - \frac{460571349685075347042722861962319}{1054711806288074294218041425034240} a^{6} - \frac{1290295246003088401280538968798887}{3164135418864222882654124275102720} a^{5} - \frac{6724380376509215430877511927809}{35157060209602476473934714167808} a^{4} + \frac{359969638347590142425193310966241}{1582067709432111441327062137551360} a^{3} + \frac{34346907564022763521370647551929}{70314120419204952947869428335616} a^{2} - \frac{89255953085812574781630074065939}{632827083772844576530824855020544} a - \frac{55828043105851691695579890511}{571659515603292300389182344192}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $11$
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:  \( 46597258073.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:Q_8$ (as 20T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 8 conjugacy class representatives for $C_5^2:Q_8$
Character table for $C_5^2:Q_8$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 10.2.2060135728811015625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ R R R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
3.8.6.1$x^{8} + 9 x^{4} + 36$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
7.8.6.1$x^{8} + 35 x^{4} + 441$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
11.4.2.2$x^{4} - 11 x^{2} + 847$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$