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Properties

Label 20.0.10675123826...000.17
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}$
Root discriminant $282.76$
Ramified primes $2, 3, 5, 7$
Class number $315842560$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 308440]$ (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![36382599176401, 0, 10539698817980, 0, 1412705580870, 0, 112358158740, 0, 5869934205, 0, 216347538, 0, 6250920, 0, 147330, 0, 2460, 0, 20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 20*x^18 + 2460*x^16 + 147330*x^14 + 6250920*x^12 + 216347538*x^10 + 5869934205*x^8 + 112358158740*x^6 + 1412705580870*x^4 + 10539698817980*x^2 + 36382599176401)
 
gp: K = bnfinit(x^20 + 20*x^18 + 2460*x^16 + 147330*x^14 + 6250920*x^12 + 216347538*x^10 + 5869934205*x^8 + 112358158740*x^6 + 1412705580870*x^4 + 10539698817980*x^2 + 36382599176401, 1)
 

Normalized defining polynomial

\( x^{20} + 20 x^{18} + 2460 x^{16} + 147330 x^{14} + 6250920 x^{12} + 216347538 x^{10} + 5869934205 x^{8} + 112358158740 x^{6} + 1412705580870 x^{4} + 10539698817980 x^{2} + 36382599176401 \)

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:  \(10675123826048640000000000000000000000000000000000=2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.76$
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 Galois and abelian over $\Q$.
Conductor:  \(4200=2^{3}\cdot 3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{4200}(1,·)$, $\chi_{4200}(3779,·)$, $\chi_{4200}(1861,·)$, $\chi_{4200}(2759,·)$, $\chi_{4200}(841,·)$, $\chi_{4200}(2701,·)$, $\chi_{4200}(3599,·)$, $\chi_{4200}(1681,·)$, $\chi_{4200}(3541,·)$, $\chi_{4200}(2521,·)$, $\chi_{4200}(3361,·)$, $\chi_{4200}(419,·)$, $\chi_{4200}(1259,·)$, $\chi_{4200}(239,·)$, $\chi_{4200}(2099,·)$, $\chi_{4200}(181,·)$, $\chi_{4200}(1079,·)$, $\chi_{4200}(2939,·)$, $\chi_{4200}(1021,·)$, $\chi_{4200}(1919,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{6} - \frac{1}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{3} + \frac{1}{9} a$, $\frac{1}{18} a^{10} + \frac{1}{9} a^{6} - \frac{1}{18} a^{4} + \frac{7}{18}$, $\frac{1}{18} a^{11} + \frac{1}{9} a^{7} - \frac{1}{18} a^{5} + \frac{7}{18} a$, $\frac{1}{378} a^{12} + \frac{61}{378} a^{6} - \frac{1}{2} a^{2} - \frac{31}{189}$, $\frac{1}{378} a^{13} + \frac{61}{378} a^{7} - \frac{1}{2} a^{3} - \frac{31}{189} a$, $\frac{1}{2646} a^{14} - \frac{1}{1323} a^{12} - \frac{1}{126} a^{10} + \frac{61}{2646} a^{8} - \frac{208}{1323} a^{6} + \frac{8}{63} a^{4} + \frac{410}{1323} a^{2} - \frac{401}{2646}$, $\frac{1}{5292} a^{15} - \frac{1}{2646} a^{13} - \frac{1}{756} a^{12} + \frac{1}{42} a^{11} - \frac{1}{36} a^{10} + \frac{61}{5292} a^{9} - \frac{1}{18} a^{8} - \frac{61}{2646} a^{7} - \frac{61}{756} a^{6} - \frac{11}{84} a^{5} - \frac{5}{36} a^{4} + \frac{646}{1323} a^{3} + \frac{5}{36} a^{2} - \frac{127}{2646} a + \frac{125}{756}$, $\frac{1}{111132} a^{16} + \frac{5}{55566} a^{14} - \frac{1}{756} a^{13} - \frac{20}{27783} a^{12} - \frac{1}{36} a^{11} - \frac{2837}{111132} a^{10} - \frac{1}{18} a^{9} + \frac{2069}{55566} a^{8} - \frac{61}{756} a^{7} - \frac{7967}{111132} a^{6} - \frac{5}{36} a^{5} + \frac{1985}{55566} a^{4} + \frac{5}{36} a^{3} - \frac{8408}{27783} a^{2} + \frac{125}{756} a + \frac{13379}{55566}$, $\frac{1}{111132} a^{17} + \frac{5}{55566} a^{15} - \frac{1}{5292} a^{14} - \frac{20}{27783} a^{13} - \frac{5}{5292} a^{12} - \frac{2837}{111132} a^{11} - \frac{1}{42} a^{10} + \frac{2069}{55566} a^{9} - \frac{61}{5292} a^{8} - \frac{7967}{111132} a^{7} - \frac{305}{5292} a^{6} + \frac{1985}{55566} a^{5} + \frac{11}{84} a^{4} - \frac{8408}{27783} a^{3} + \frac{1385}{5292} a^{2} + \frac{13379}{55566} a + \frac{172}{1323}$, $\frac{1}{15542397477957632445733898364353462750724} a^{18} + \frac{12562276414547444732189679204933865}{7771198738978816222866949182176731375362} a^{16} + \frac{714940409071156653189645702980340941}{3885599369489408111433474591088365687681} a^{14} - \frac{1}{756} a^{13} + \frac{4279125915177259543540314673463200868}{3885599369489408111433474591088365687681} a^{12} - \frac{296596858184349781223309416524551891863}{15542397477957632445733898364353462750724} a^{10} + \frac{522172289420421171536352894391078598591}{15542397477957632445733898364353462750724} a^{8} - \frac{61}{756} a^{7} + \frac{2303133997193645967456224293290470079647}{15542397477957632445733898364353462750724} a^{6} + \frac{2049853365290253885309449880653803083107}{15542397477957632445733898364353462750724} a^{4} - \frac{1}{4} a^{3} - \frac{5481778782867424382168031204304369625299}{15542397477957632445733898364353462750724} a^{2} + \frac{31}{378} a - \frac{1033021233829272500757335604080669767331}{15542397477957632445733898364353462750724}$, $\frac{1}{93748617565147369428545282420208852266354272476} a^{19} + \frac{25741216807705003949525126651665320004657}{46874308782573684714272641210104426133177136238} a^{17} + \frac{1386706978292174946362630298387412866705599}{23437154391286842357136320605052213066588568119} a^{15} - \frac{1}{5292} a^{14} + \frac{13605165095990723500561392703239409653493607}{23437154391286842357136320605052213066588568119} a^{13} + \frac{1}{2646} a^{12} - \frac{1989310095602272569497893298348911690119168697}{93748617565147369428545282420208852266354272476} a^{11} - \frac{1}{42} a^{10} - \frac{4028552823129189225770483833079064538738381093}{93748617565147369428545282420208852266354272476} a^{9} - \frac{61}{5292} a^{8} - \frac{9687183749615675992077353164100164035475491397}{93748617565147369428545282420208852266354272476} a^{7} + \frac{61}{2646} a^{6} - \frac{3934690281291049316935765321211838601601169727}{93748617565147369428545282420208852266354272476} a^{5} + \frac{11}{84} a^{4} + \frac{747740727687982750819516927437885108287654957}{93748617565147369428545282420208852266354272476} a^{3} - \frac{646}{1323} a^{2} + \frac{6457888763284271965600672780143974106872182903}{93748617565147369428545282420208852266354272476} a + \frac{127}{2646}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{308440}$, which has order $315842560$ (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:  \( 180801817.57689384 \) (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

\(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-14}, \sqrt{15})\), 5.5.390625.1, 10.0.102102525000000000000000.1, 10.0.84035000000000000000.3, 10.10.189843750000000000.1

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 R R R ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$5$5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$