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Properties

Label 20.0.67955065915...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $438.14$
Ramified primes $2, 3, 5, 41$
Class number $1335217152$ (GRH)
Class group $[2, 2, 2, 2, 2, 4, 10431384]$ (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![136161, 0, 10045656, 0, 63170299, 0, 149985544, 0, 167015755, 0, 88124416, 0, 19048600, 0, 1261816, 0, 32185, 0, 328, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 328*x^18 + 32185*x^16 + 1261816*x^14 + 19048600*x^12 + 88124416*x^10 + 167015755*x^8 + 149985544*x^6 + 63170299*x^4 + 10045656*x^2 + 136161)
 
gp: K = bnfinit(x^20 + 328*x^18 + 32185*x^16 + 1261816*x^14 + 19048600*x^12 + 88124416*x^10 + 167015755*x^8 + 149985544*x^6 + 63170299*x^4 + 10045656*x^2 + 136161, 1)
 

Normalized defining polynomial

\( x^{20} + 328 x^{18} + 32185 x^{16} + 1261816 x^{14} + 19048600 x^{12} + 88124416 x^{10} + 167015755 x^{8} + 149985544 x^{6} + 63170299 x^{4} + 10045656 x^{2} + 136161 \)

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:  \(67955065915961530473358856164812748971048960000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.14$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(4099,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(2519,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(3139,·)$, $\chi_{4920}(3221,·)$, $\chi_{4920}(1559,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(4459,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(4699,·)$, $\chi_{4920}(619,·)$, $\chi_{4920}(4781,·)$, $\chi_{4920}(1199,·)$, $\chi_{4920}(119,·)$, $\chi_{4920}(4541,·)$, $\chi_{4920}(959,·)$, $\chi_{4920}(701,·)$, $\chi_{4920}(4181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{3}$, $\frac{1}{369} a^{10} - \frac{2}{9} a^{4}$, $\frac{1}{1107} a^{11} + \frac{1}{27} a^{9} - \frac{1}{9} a^{7} + \frac{4}{27} a^{5} + \frac{5}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{1107} a^{12} - \frac{1}{1107} a^{10} + \frac{4}{27} a^{6} + \frac{8}{27} a^{4} + \frac{2}{9} a^{2}$, $\frac{1}{1107} a^{13} + \frac{1}{27} a^{9} + \frac{1}{27} a^{7} + \frac{1}{9} a^{5} + \frac{2}{27} a^{3}$, $\frac{1}{1107} a^{14} - \frac{1}{1107} a^{10} + \frac{1}{27} a^{8} + \frac{1}{9} a^{6} + \frac{5}{27} a^{4}$, $\frac{1}{1107} a^{15} - \frac{1}{27} a^{9} - \frac{1}{27} a^{3}$, $\frac{1}{727299} a^{16} + \frac{103}{242433} a^{14} + \frac{95}{242433} a^{12} - \frac{407}{727299} a^{10} - \frac{278}{5913} a^{8} - \frac{337}{5913} a^{6} + \frac{4502}{17739} a^{4} - \frac{250}{1971} a^{2} + \frac{74}{219}$, $\frac{1}{727299} a^{17} + \frac{103}{242433} a^{15} + \frac{95}{242433} a^{13} + \frac{250}{727299} a^{11} - \frac{59}{5913} a^{9} + \frac{977}{5913} a^{7} + \frac{1217}{17739} a^{5} - \frac{542}{1971} a^{3} + \frac{1}{219} a$, $\frac{1}{5752178129065335499953} a^{18} - \frac{3618005148006637}{5752178129065335499953} a^{16} - \frac{534945256836417770}{1917392709688445166651} a^{14} + \frac{1660824815484221668}{5752178129065335499953} a^{12} + \frac{6818092769275704371}{5752178129065335499953} a^{10} - \frac{700638448418881337}{46765675846059638211} a^{8} - \frac{10383503412337822723}{140297027538178914633} a^{6} - \frac{347867359208496167}{1921877089564094721} a^{4} + \frac{7118216030273115877}{15588558615353212737} a^{2} - \frac{706316883651633050}{1732062068372579193}$, $\frac{1}{5752178129065335499953} a^{19} - \frac{3618005148006637}{5752178129065335499953} a^{17} - \frac{534945256836417770}{1917392709688445166651} a^{15} + \frac{1660824815484221668}{5752178129065335499953} a^{13} + \frac{1621906564157966792}{5752178129065335499953} a^{11} - \frac{2432700516791460530}{46765675846059638211} a^{9} + \frac{5205055203015390014}{140297027538178914633} a^{7} + \frac{8035805525595448}{1921877089564094721} a^{5} - \frac{6160926493916657936}{15588558615353212737} a^{3} - \frac{706316883651633050}{1732062068372579193} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{10431384}$, which has order $1335217152$ (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:  \( 41502978620.14737 \) (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

\(\Q(\sqrt{15}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{-246}) \), \(\Q(\sqrt{15}, \sqrt{-246})\), 5.5.2825761.1, 10.10.6209077858164489600000.1, 10.0.33523910081941606400000.1, 10.0.2606819247971779313664.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$5$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
41Data not computed