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Properties

Label 20.4.127...976.1
Degree $20$
Signature $[4, 8]$
Discriminant $1.273\times 10^{26}$
Root discriminant \(20.20\)
Ramified primes $2,3$
Class number $1$
Class group trivial
Galois group $A_6$ (as 20T89)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1)
 
gp: K = bnfinit(y^20 - 2*y^18 + 9*y^16 + 12*y^14 - 6*y^12 + 30*y^8 - 60*y^6 + 45*y^4 - 14*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1)
 

\( x^{20} - 2x^{18} + 9x^{16} + 12x^{14} - 6x^{12} + 30x^{8} - 60x^{6} + 45x^{4} - 14x^{2} + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $20$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(127338577759142414150270976\) \(\medspace = 2^{36}\cdot 3^{32}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(20.20\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{13/6}3^{16/9}\approx 31.655357399783927$
Ramified primes:   \(2\), \(3\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{2}-\frac{1}{2}a+\frac{3}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a^{3}+\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{16}a^{12}+\frac{3}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{7}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a-\frac{1}{16}$, $\frac{1}{16}a^{13}+\frac{3}{16}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{7}{16}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}+\frac{1}{4}a^{2}-\frac{1}{16}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}-\frac{1}{4}a^{7}+\frac{1}{16}a^{6}+\frac{1}{4}a^{5}-\frac{1}{8}a^{4}+\frac{1}{4}a^{3}-\frac{5}{16}a^{2}-\frac{1}{4}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{11}+\frac{1}{8}a^{9}+\frac{1}{16}a^{7}-\frac{1}{4}a^{6}-\frac{1}{8}a^{5}+\frac{1}{4}a^{4}-\frac{5}{16}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{32}a^{16}-\frac{1}{16}a^{8}-\frac{1}{8}a^{6}+\frac{1}{4}a^{4}-\frac{3}{8}a^{2}-\frac{1}{2}a-\frac{15}{32}$, $\frac{1}{32}a^{17}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}+\frac{1}{4}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}-\frac{15}{32}a$, $\frac{1}{96}a^{18}-\frac{1}{16}a^{10}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}-\frac{1}{2}a^{3}-\frac{9}{32}a^{2}-\frac{1}{2}a-\frac{11}{24}$, $\frac{1}{96}a^{19}-\frac{1}{16}a^{11}-\frac{1}{4}a^{7}-\frac{3}{8}a^{5}-\frac{9}{32}a^{3}-\frac{11}{24}a-\frac{1}{2}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $a$, $\frac{15}{32}a^{19}-\frac{31}{48}a^{18}-\frac{35}{32}a^{17}+\frac{17}{16}a^{16}+\frac{35}{8}a^{15}-\frac{87}{16}a^{14}+\frac{71}{16}a^{13}-\frac{155}{16}a^{12}-\frac{95}{16}a^{11}+\frac{7}{16}a^{10}-\frac{7}{4}a^{9}+\frac{1}{16}a^{8}+\frac{109}{8}a^{7}-\frac{313}{16}a^{6}-\frac{527}{16}a^{5}+\frac{511}{16}a^{4}+\frac{815}{32}a^{3}-\frac{137}{8}a^{2}-\frac{245}{32}a+\frac{67}{24}$, $\frac{9}{16}a^{19}+\frac{3}{16}a^{18}-\frac{17}{16}a^{17}-\frac{1}{4}a^{16}+5a^{15}+\frac{23}{16}a^{14}+\frac{29}{4}a^{13}+\frac{53}{16}a^{12}-\frac{17}{8}a^{11}+\frac{7}{16}a^{10}+\frac{7}{8}a^{9}-\frac{19}{16}a^{8}+\frac{71}{4}a^{7}+\frac{69}{16}a^{6}-31a^{5}-\frac{125}{16}a^{4}+\frac{389}{16}a^{3}+\frac{9}{8}a^{2}-\frac{105}{16}a+\frac{15}{16}$, $\frac{17}{16}a^{19}+\frac{1}{2}a^{18}-\frac{27}{16}a^{17}-\frac{3}{4}a^{16}+\frac{143}{16}a^{15}+\frac{33}{8}a^{14}+\frac{261}{16}a^{13}+\frac{129}{16}a^{12}+\frac{15}{16}a^{11}+a^{10}+\frac{23}{16}a^{9}+\frac{9}{16}a^{8}+\frac{517}{16}a^{7}+15a^{6}-\frac{801}{16}a^{5}-\frac{367}{16}a^{4}+\frac{117}{4}a^{3}+\frac{89}{8}a^{2}-\frac{13}{2}a-\frac{27}{16}$, $\frac{11}{24}a^{19}-\frac{9}{32}a^{18}-\frac{23}{32}a^{17}+\frac{7}{16}a^{16}+\frac{15}{4}a^{15}-\frac{19}{8}a^{14}+\frac{115}{16}a^{13}-\frac{35}{8}a^{12}-\frac{1}{8}a^{11}-\frac{9}{16}a^{10}-\frac{11}{8}a^{9}-\frac{7}{8}a^{8}+\frac{51}{4}a^{7}-\frac{71}{8}a^{6}-\frac{357}{16}a^{5}+13a^{4}+\frac{37}{4}a^{3}-\frac{229}{32}a^{2}-\frac{43}{96}a+\frac{29}{16}$, $\frac{31}{48}a^{19}+\frac{61}{96}a^{18}-\frac{17}{16}a^{17}-\frac{29}{32}a^{16}+\frac{87}{16}a^{15}+\frac{83}{16}a^{14}+\frac{155}{16}a^{13}+\frac{85}{8}a^{12}-\frac{7}{16}a^{11}+\frac{17}{8}a^{10}-\frac{1}{16}a^{9}+\frac{19}{16}a^{8}+\frac{313}{16}a^{7}+\frac{309}{16}a^{6}-\frac{511}{16}a^{5}-\frac{221}{8}a^{4}+\frac{137}{8}a^{3}+\frac{381}{32}a^{2}-\frac{67}{24}a-\frac{107}{96}$, $\frac{65}{96}a^{19}-\frac{15}{32}a^{18}-\frac{43}{32}a^{17}+\frac{17}{32}a^{16}+\frac{95}{16}a^{15}-\frac{59}{16}a^{14}+\frac{67}{8}a^{13}-\frac{71}{8}a^{12}-5a^{11}-\frac{35}{8}a^{10}-\frac{41}{16}a^{9}-\frac{35}{16}a^{8}+\frac{305}{16}a^{7}-\frac{241}{16}a^{6}-\frac{323}{8}a^{5}+\frac{125}{8}a^{4}+\frac{837}{32}a^{3}-\frac{173}{32}a^{2}-\frac{409}{96}a+\frac{29}{32}$, $\frac{65}{96}a^{19}+\frac{15}{32}a^{18}-\frac{43}{32}a^{17}-\frac{17}{32}a^{16}+\frac{95}{16}a^{15}+\frac{59}{16}a^{14}+\frac{67}{8}a^{13}+\frac{71}{8}a^{12}-5a^{11}+\frac{35}{8}a^{10}-\frac{41}{16}a^{9}+\frac{35}{16}a^{8}+\frac{305}{16}a^{7}+\frac{241}{16}a^{6}-\frac{323}{8}a^{5}-\frac{125}{8}a^{4}+\frac{837}{32}a^{3}+\frac{173}{32}a^{2}-\frac{409}{96}a-\frac{29}{32}$, $\frac{1}{3}a^{19}+\frac{9}{32}a^{18}-\frac{19}{32}a^{17}-\frac{1}{2}a^{16}+\frac{45}{16}a^{15}+\frac{5}{2}a^{14}+\frac{37}{8}a^{13}+\frac{31}{8}a^{12}-\frac{21}{16}a^{11}-\frac{5}{16}a^{10}-\frac{29}{16}a^{9}+\frac{7}{4}a^{8}+\frac{135}{16}a^{7}+\frac{39}{4}a^{6}-\frac{69}{4}a^{5}-\frac{63}{4}a^{4}+\frac{181}{16}a^{3}+\frac{337}{32}a^{2}-\frac{253}{96}a-\frac{19}{8}$, $\frac{25}{48}a^{19}+\frac{1}{6}a^{18}-a^{17}-\frac{1}{4}a^{16}+\frac{37}{8}a^{15}+\frac{11}{8}a^{14}+\frac{53}{8}a^{13}+\frac{43}{16}a^{12}-\frac{5}{2}a^{11}+\frac{3}{8}a^{10}+\frac{3}{8}a^{9}+\frac{1}{16}a^{8}+\frac{129}{8}a^{7}+\frac{11}{2}a^{6}-\frac{237}{8}a^{5}-\frac{121}{16}a^{4}+\frac{353}{16}a^{3}+4a^{2}-\frac{145}{24}a-\frac{25}{48}$, $\frac{25}{48}a^{19}-\frac{1}{6}a^{18}-a^{17}+\frac{1}{4}a^{16}+\frac{37}{8}a^{15}-\frac{11}{8}a^{14}+\frac{53}{8}a^{13}-\frac{43}{16}a^{12}-\frac{5}{2}a^{11}-\frac{3}{8}a^{10}+\frac{3}{8}a^{9}-\frac{1}{16}a^{8}+\frac{129}{8}a^{7}-\frac{11}{2}a^{6}-\frac{237}{8}a^{5}+\frac{121}{16}a^{4}+\frac{353}{16}a^{3}-4a^{2}-\frac{145}{24}a+\frac{25}{48}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 616938.09204 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 616938.09204 \cdot 1}{2\cdot\sqrt{127338577759142414150270976}}\cr\approx \mathstrut & 1.0624060188 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^18 + 9*x^16 + 12*x^14 - 6*x^12 + 30*x^8 - 60*x^6 + 45*x^4 - 14*x^2 + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$A_6$ (as 20T89):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 360
The 7 conjugacy class representatives for $A_6$
Character table for $A_6$

Intermediate fields

10.2.11284439629824.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 6 siblings: 6.2.1679616.2, 6.2.6718464.1
Degree 10 sibling: 10.2.11284439629824.1
Degree 15 siblings: deg 15, 15.3.18953525353286467584.1
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.2.1679616.2

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 ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.5.0.1}{5} }^{4}$ ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.5.0.1}{5} }^{4}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.5.0.1}{5} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.4.8.7$x^{4} + 4 x^{2} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.34$x^{12} + 4 x^{11} + 12 x^{10} + 24 x^{9} + 44 x^{8} + 32 x^{7} + 48 x^{6} + 16 x^{5} + 68 x^{4} + 32 x^{3} + 24 x^{2} + 28$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
\(3\) Copy content Toggle raw display 3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $18$$9$$2$$32$