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Properties

Label 20.0.43774679639...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}$
Root discriminant $38.20$
Ramified primes $2, 3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (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![21256, -30600, 28696, 14640, -19272, -4530, 36366, -13752, 2250, 9534, -2283, 1776, 1953, -324, 798, -84, 168, -12, 19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 19*x^18 - 12*x^17 + 168*x^16 - 84*x^15 + 798*x^14 - 324*x^13 + 1953*x^12 + 1776*x^11 - 2283*x^10 + 9534*x^9 + 2250*x^8 - 13752*x^7 + 36366*x^6 - 4530*x^5 - 19272*x^4 + 14640*x^3 + 28696*x^2 - 30600*x + 21256)
 
gp: K = bnfinit(x^20 + 19*x^18 - 12*x^17 + 168*x^16 - 84*x^15 + 798*x^14 - 324*x^13 + 1953*x^12 + 1776*x^11 - 2283*x^10 + 9534*x^9 + 2250*x^8 - 13752*x^7 + 36366*x^6 - 4530*x^5 - 19272*x^4 + 14640*x^3 + 28696*x^2 - 30600*x + 21256, 1)
 

Normalized defining polynomial

\( x^{20} + 19 x^{18} - 12 x^{17} + 168 x^{16} - 84 x^{15} + 798 x^{14} - 324 x^{13} + 1953 x^{12} + 1776 x^{11} - 2283 x^{10} + 9534 x^{9} + 2250 x^{8} - 13752 x^{7} + 36366 x^{6} - 4530 x^{5} - 19272 x^{4} + 14640 x^{3} + 28696 x^{2} - 30600 x + 21256 \)

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:  \(43774679639190704400000000000000=2^{16}\cdot 3^{18}\cdot 5^{14}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.20$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{12} - \frac{1}{6} a^{9} - \frac{1}{18} a^{8} + \frac{1}{3} a^{7} - \frac{1}{18} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{13} + \frac{1}{9} a^{9} - \frac{7}{18} a^{7} - \frac{1}{2} a^{5} - \frac{4}{9} a^{3} - \frac{4}{9} a$, $\frac{1}{180} a^{16} + \frac{1}{45} a^{15} + \frac{1}{45} a^{14} + \frac{1}{45} a^{13} - \frac{1}{30} a^{11} - \frac{1}{45} a^{10} - \frac{7}{45} a^{9} + \frac{11}{180} a^{8} - \frac{11}{90} a^{7} - \frac{1}{2} a^{6} - \frac{7}{30} a^{5} + \frac{29}{90} a^{4} - \frac{43}{90} a^{3} + \frac{4}{45} a^{2} + \frac{16}{45} a + \frac{1}{5}$, $\frac{1}{180} a^{17} - \frac{1}{90} a^{15} - \frac{1}{90} a^{14} - \frac{1}{30} a^{13} + \frac{1}{45} a^{12} - \frac{1}{18} a^{11} - \frac{1}{15} a^{10} + \frac{23}{180} a^{9} - \frac{4}{45} a^{8} + \frac{1}{10} a^{7} + \frac{2}{45} a^{6} + \frac{4}{45} a^{5} + \frac{7}{30} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a + \frac{19}{45}$, $\frac{1}{180} a^{18} - \frac{1}{45} a^{15} + \frac{1}{90} a^{14} + \frac{1}{90} a^{13} - \frac{1}{18} a^{12} + \frac{1}{30} a^{11} - \frac{1}{12} a^{10} + \frac{7}{45} a^{9} - \frac{1}{9} a^{8} - \frac{14}{45} a^{7} - \frac{7}{90} a^{6} - \frac{1}{15} a^{5} - \frac{2}{15} a^{4} + \frac{17}{45} a^{3} + \frac{8}{45} a^{2} - \frac{19}{45} a - \frac{4}{15}$, $\frac{1}{4795352595146701802042309082128697540} a^{19} - \frac{826550666120309057268892094644649}{1598450865048900600680769694042899180} a^{18} + \frac{11020850139501568237295464193487007}{4795352595146701802042309082128697540} a^{17} - \frac{95711821387394973027942197386064}{79922543252445030034038484702144959} a^{16} + \frac{28145125666290803363661118085807743}{1198838148786675450510577270532174385} a^{15} + \frac{609013549786854758671120391158297}{159845086504890060068076969404289918} a^{14} - \frac{1768539363900011204534780494993111}{159845086504890060068076969404289918} a^{13} - \frac{11158861941566090963623393364100017}{266408477508150100113461615673816530} a^{12} - \frac{97109137033656645889926882696050311}{4795352595146701802042309082128697540} a^{11} + \frac{101286620785740312093929880061183631}{1598450865048900600680769694042899180} a^{10} - \frac{357335338744510255657086624787670677}{4795352595146701802042309082128697540} a^{9} + \frac{34656746161175425570259526325699987}{266408477508150100113461615673816530} a^{8} - \frac{29494599879282494936372539008298597}{399612716262225150170192423510724795} a^{7} - \frac{8075782414386636481576471630886331}{266408477508150100113461615673816530} a^{6} - \frac{826739584791236134525501808666168923}{2397676297573350901021154541064348770} a^{5} - \frac{30875748524228899749180688500535401}{266408477508150100113461615673816530} a^{4} - \frac{263605769492633668356490273059821312}{1198838148786675450510577270532174385} a^{3} - \frac{104373559028217412996411649923863688}{399612716262225150170192423510724795} a^{2} - \frac{555414722897366293271502385445699}{4523917542591228115134253851064809} a + \frac{30300829883214177347298868009020646}{399612716262225150170192423510724795}$

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:  $9$
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:  \( 239722179.86484048 \) (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-7}, \sqrt{-15})\), 5.1.162000.1, 10.0.393660000000.1, 10.0.441082908000000.2, 10.2.6616243620000000.2

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 R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$$[\ ]_{4}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$