Properties

Label 20.0.369...048.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.691\times 10^{20}$
Root discriminant $10.67$
Ramified primes $2, 3, 157$
Class number $1$
Class group trivial
Galois group $D_5\wr C_2$ (as 20T48)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 26*x^18 - 64*x^17 + 116*x^16 - 153*x^15 + 128*x^14 - 4*x^13 - 199*x^12 + 398*x^11 - 478*x^10 + 372*x^9 - 118*x^8 - 158*x^7 + 328*x^6 - 344*x^5 + 249*x^4 - 129*x^3 + 46*x^2 - 10*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 46, -129, 249, -344, 328, -158, -118, 372, -478, 398, -199, -4, 128, -153, 116, -64, 26, -7, 1]);
 

\( x^{20} - 7 x^{19} + 26 x^{18} - 64 x^{17} + 116 x^{16} - 153 x^{15} + 128 x^{14} - 4 x^{13} - 199 x^{12} + 398 x^{11} - 478 x^{10} + 372 x^{9} - 118 x^{8} - 158 x^{7} + 328 x^{6} - 344 x^{5} + 249 x^{4} - 129 x^{3} + 46 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(369139687194512130048\)\(\medspace = 2^{16}\cdot 3^{10}\cdot 157^{5}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $10.67$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 157$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31} a^{18} - \frac{1}{31} a^{17} - \frac{5}{31} a^{16} - \frac{7}{31} a^{15} + \frac{13}{31} a^{14} + \frac{7}{31} a^{13} + \frac{7}{31} a^{11} - \frac{2}{31} a^{10} - \frac{6}{31} a^{9} + \frac{1}{31} a^{8} + \frac{1}{31} a^{7} - \frac{13}{31} a^{6} - \frac{13}{31} a^{5} - \frac{14}{31} a^{4} - \frac{10}{31} a^{3} + \frac{12}{31} a^{2} + \frac{7}{31} a + \frac{5}{31}$, $\frac{1}{5053} a^{19} - \frac{54}{5053} a^{18} - \frac{696}{5053} a^{17} + \frac{537}{5053} a^{16} + \frac{1283}{5053} a^{15} - \frac{73}{163} a^{14} - \frac{1580}{5053} a^{13} + \frac{906}{5053} a^{12} - \frac{2357}{5053} a^{11} + \frac{2456}{5053} a^{10} + \frac{1776}{5053} a^{9} + \frac{2149}{5053} a^{8} + \frac{1732}{5053} a^{7} + \frac{1079}{5053} a^{6} + \frac{2101}{5053} a^{5} - \frac{1128}{5053} a^{4} - \frac{2155}{5053} a^{3} + \frac{2378}{5053} a^{2} + \frac{750}{5053} a - \frac{2497}{5053}$

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

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{92}{31} a^{19} - \frac{627}{31} a^{18} + \frac{2307}{31} a^{17} - \frac{5626}{31} a^{16} + \frac{10118}{31} a^{15} - \frac{13131}{31} a^{14} + \frac{10639}{31} a^{13} + \frac{551}{31} a^{12} - \frac{18282}{31} a^{11} + \frac{35176}{31} a^{10} - \frac{40966}{31} a^{9} + \frac{30588}{31} a^{8} - \frac{7714}{31} a^{7} - \frac{15910}{31} a^{6} + \frac{29382}{31} a^{5} - \frac{29235}{31} a^{4} + \frac{20032}{31} a^{3} - \frac{9527}{31} a^{2} + \frac{2853}{31} a - \frac{381}{31} \) (order $6$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 182.345344476 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{10}\cdot 182.345344476 \cdot 1}{6\sqrt{369139687194512130048}}\approx 0.151686434459$

Galois group

$D_5\wr C_2$ (as 20T48):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 4.0.1413.1, 10.0.1533364992.1 x5

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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}$
157Data not computed