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Properties

Label 20.0.20439072898...1936.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 11^{16}\cdot 23^{10}$
Root discriminant $92.37$
Ramified primes $2, 11, 23$
Class number $314880$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 1230]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2836553039, -1232813294, 2285123581, -807249866, 799590552, -239987380, 167779661, -44848672, 24527620, -5992422, 2693016, -611754, 234490, -49894, 16372, -3052, 854, -140, 37, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 37*x^18 - 140*x^17 + 854*x^16 - 3052*x^15 + 16372*x^14 - 49894*x^13 + 234490*x^12 - 611754*x^11 + 2693016*x^10 - 5992422*x^9 + 24527620*x^8 - 44848672*x^7 + 167779661*x^6 - 239987380*x^5 + 799590552*x^4 - 807249866*x^3 + 2285123581*x^2 - 1232813294*x + 2836553039)
 
gp: K = bnfinit(x^20 - 6*x^19 + 37*x^18 - 140*x^17 + 854*x^16 - 3052*x^15 + 16372*x^14 - 49894*x^13 + 234490*x^12 - 611754*x^11 + 2693016*x^10 - 5992422*x^9 + 24527620*x^8 - 44848672*x^7 + 167779661*x^6 - 239987380*x^5 + 799590552*x^4 - 807249866*x^3 + 2285123581*x^2 - 1232813294*x + 2836553039, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 37 x^{18} - 140 x^{17} + 854 x^{16} - 3052 x^{15} + 16372 x^{14} - 49894 x^{13} + 234490 x^{12} - 611754 x^{11} + 2693016 x^{10} - 5992422 x^{9} + 24527620 x^{8} - 44848672 x^{7} + 167779661 x^{6} - 239987380 x^{5} + 799590552 x^{4} - 807249866 x^{3} + 2285123581 x^{2} - 1232813294 x + 2836553039 \)

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:  \(2043907289848891215377420272534275751936=2^{30}\cdot 11^{16}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.37$
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:  $2, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2024=2^{3}\cdot 11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{2024}(1,·)$, $\chi_{2024}(1609,·)$, $\chi_{2024}(45,·)$, $\chi_{2024}(1105,·)$, $\chi_{2024}(597,·)$, $\chi_{2024}(137,·)$, $\chi_{2024}(1241,·)$, $\chi_{2024}(1565,·)$, $\chi_{2024}(93,·)$, $\chi_{2024}(1057,·)$, $\chi_{2024}(229,·)$, $\chi_{2024}(1885,·)$, $\chi_{2024}(553,·)$, $\chi_{2024}(1197,·)$, $\chi_{2024}(829,·)$, $\chi_{2024}(1841,·)$, $\chi_{2024}(1013,·)$, $\chi_{2024}(873,·)$, $\chi_{2024}(185,·)$, $\chi_{2024}(1149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} + \frac{11}{31} a^{14} - \frac{5}{31} a^{13} + \frac{4}{31} a^{12} - \frac{4}{31} a^{11} + \frac{12}{31} a^{10} - \frac{15}{31} a^{9} - \frac{8}{31} a^{8} - \frac{2}{31} a^{7} - \frac{15}{31} a^{6} + \frac{10}{31} a^{5} + \frac{5}{31} a^{4} - \frac{12}{31} a^{3} - \frac{11}{31} a^{2} - \frac{6}{31} a - \frac{8}{31}$, $\frac{1}{31} a^{16} - \frac{2}{31} a^{14} - \frac{3}{31} a^{13} + \frac{14}{31} a^{12} - \frac{6}{31} a^{11} + \frac{8}{31} a^{10} + \frac{2}{31} a^{9} - \frac{7}{31} a^{8} + \frac{7}{31} a^{7} - \frac{11}{31} a^{6} - \frac{12}{31} a^{5} - \frac{5}{31} a^{4} - \frac{3}{31} a^{3} - \frac{9}{31} a^{2} - \frac{4}{31} a - \frac{5}{31}$, $\frac{1}{31} a^{17} - \frac{12}{31} a^{14} + \frac{4}{31} a^{13} + \frac{2}{31} a^{12} - \frac{5}{31} a^{10} - \frac{6}{31} a^{9} - \frac{9}{31} a^{8} - \frac{15}{31} a^{7} - \frac{11}{31} a^{6} + \frac{15}{31} a^{5} + \frac{7}{31} a^{4} - \frac{2}{31} a^{3} + \frac{5}{31} a^{2} + \frac{14}{31} a + \frac{15}{31}$, $\frac{1}{2728588336631173612121} a^{18} - \frac{16852172048705003613}{2728588336631173612121} a^{17} + \frac{37565472888323780773}{2728588336631173612121} a^{16} - \frac{35845049306058840552}{2728588336631173612121} a^{15} + \frac{1343676905472448207122}{2728588336631173612121} a^{14} + \frac{1132286516577722990683}{2728588336631173612121} a^{13} + \frac{1265634734439992146027}{2728588336631173612121} a^{12} - \frac{68541018748580671237}{2728588336631173612121} a^{11} - \frac{387855770843876230671}{2728588336631173612121} a^{10} - \frac{1098956761960223746603}{2728588336631173612121} a^{9} - \frac{14198909625539096905}{30658295917204197889} a^{8} - \frac{126785732394156998077}{2728588336631173612121} a^{7} - \frac{29346462892317900678}{2728588336631173612121} a^{6} + \frac{467668717281068051287}{2728588336631173612121} a^{5} - \frac{21309251912438617051}{118634275505703200527} a^{4} + \frac{16400604578904153827}{88018978601005600391} a^{3} - \frac{94419337941638287797}{2728588336631173612121} a^{2} + \frac{20256354606190244952}{118634275505703200527} a + \frac{56881485813939614697}{118634275505703200527}$, $\frac{1}{8223358027319367589563728592168378410549564721221801} a^{19} - \frac{1122223020604695139775918220373}{8223358027319367589563728592168378410549564721221801} a^{18} - \frac{18110578067254329859210369082245618063576372590121}{8223358027319367589563728592168378410549564721221801} a^{17} + \frac{78712521715212402276251093828913825915120513354004}{8223358027319367589563728592168378410549564721221801} a^{16} + \frac{94560094271293763214058129775420858882074001415448}{8223358027319367589563728592168378410549564721221801} a^{15} + \frac{74528622040957987000342286100396434808816886288790}{265269613784495728695604148134463819695147249071671} a^{14} + \frac{2569345683618720394366419505963597313084687257412554}{8223358027319367589563728592168378410549564721221801} a^{13} + \frac{1579318116625783093291756221615343863048637958301205}{8223358027319367589563728592168378410549564721221801} a^{12} - \frac{1404705049705524063462195567951607792556736396216438}{8223358027319367589563728592168378410549564721221801} a^{11} + \frac{1008346827547813597398058936283268440366239414274707}{8223358027319367589563728592168378410549564721221801} a^{10} - \frac{3034361922944931634099196813542748632043806427295612}{8223358027319367589563728592168378410549564721221801} a^{9} + \frac{1586528657440947855772216614105485960542333892897724}{8223358027319367589563728592168378410549564721221801} a^{8} - \frac{3786853334339520604125116956504133850077365786623987}{8223358027319367589563728592168378410549564721221801} a^{7} - \frac{2769920202138631010866241586625090156555156928552312}{8223358027319367589563728592168378410549564721221801} a^{6} - \frac{1468450492051288269641903803176244658264139555651653}{8223358027319367589563728592168378410549564721221801} a^{5} + \frac{1942185190403479547669571719472376130950180765813333}{8223358027319367589563728592168378410549564721221801} a^{4} - \frac{340493492235624634548843941769627979336036343051419}{8223358027319367589563728592168378410549564721221801} a^{3} + \frac{473360203013255036534149123286346106594052539559532}{8223358027319367589563728592168378410549564721221801} a^{2} - \frac{173571463485318904241975920333710156539583677702636}{357537305535624677807118634442103409154328900922687} a - \frac{67686603715581536536980198175125101601150412416899}{357537305535624677807118634442103409154328900922687}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{1230}$, which has order $314880$ (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:  \( 530208.250733 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-46}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-23})\), \(\Q(\zeta_{11})^+\), 10.0.45209592896296812544.1, 10.0.1379687283212183.1, 10.10.7024111812608.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.1$x^{10} + 2 x^{8} - 4 x^{6} + 16 x^{2} - 32$$2$$5$$15$$C_{10}$$[3]^{5}$
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$