Properties

Label 20.0.250...256.1
Degree $20$
Signature $[0, 10]$
Discriminant $2.502\times 10^{24}$
Root discriminant \(16.59\)
Ramified primes $2,17$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 20T31)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4)
 
gp: K = bnfinit(y^20 - 2*y^19 + 2*y^18 + 2*y^17 - 5*y^16 + 4*y^15 + 4*y^14 - 12*y^13 + 7*y^12 + 2*y^11 - 14*y^10 + 6*y^9 + 5*y^8 - 12*y^7 + 8*y^6 + 4*y^5 - 4*y^4 + 8*y^3 + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4)
 

\( x^{20} - 2 x^{19} + 2 x^{18} + 2 x^{17} - 5 x^{16} + 4 x^{15} + 4 x^{14} - 12 x^{13} + 7 x^{12} + 2 x^{11} - 14 x^{10} + 6 x^{9} + 5 x^{8} - 12 x^{7} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 8 x^{3} + 4 \) 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:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2502343454824177132896256\) \(\medspace = 2^{32}\cdot 17^{12}\) 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:  \(16.59\)
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))
 
Ramified primes:   \(2\), \(17\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{14}+\frac{1}{4}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{15}+\frac{1}{4}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{8}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{9}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{10}$, $\frac{1}{164}a^{19}+\frac{1}{41}a^{18}-\frac{15}{164}a^{17}-\frac{3}{82}a^{16}+\frac{1}{41}a^{14}-\frac{13}{164}a^{13}-\frac{2}{41}a^{12}-\frac{1}{4}a^{11}+\frac{1}{82}a^{10}+\frac{39}{164}a^{9}-\frac{3}{82}a^{8}+\frac{5}{82}a^{7}-\frac{17}{82}a^{6}-\frac{73}{164}a^{5}-\frac{6}{41}a^{4}-\frac{33}{82}a^{3}-\frac{15}{41}a^{2}+\frac{25}{82}a-\frac{7}{41}$ 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:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{11}{41} a^{19} - \frac{111}{164} a^{18} + \frac{119}{164} a^{17} + \frac{23}{164} a^{16} - \frac{5}{4} a^{15} + \frac{44}{41} a^{14} + \frac{21}{41} a^{13} - \frac{129}{41} a^{12} + \frac{5}{2} a^{11} - \frac{35}{164} a^{10} - \frac{375}{164} a^{9} + \frac{351}{164} a^{8} + \frac{317}{164} a^{7} - \frac{215}{82} a^{6} + \frac{99}{41} a^{5} + \frac{5}{82} a^{4} - \frac{99}{82} a^{3} + \frac{37}{41} a^{2} - \frac{24}{41} a - \frac{21}{41} \)  (order $4$) 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:   $\frac{1}{164}a^{19}+\frac{1}{41}a^{18}+\frac{13}{82}a^{17}-\frac{47}{164}a^{16}+\frac{1}{4}a^{15}+\frac{43}{82}a^{14}-\frac{34}{41}a^{13}+\frac{37}{82}a^{12}+\frac{3}{4}a^{11}-\frac{163}{82}a^{10}+\frac{20}{41}a^{9}+\frac{117}{164}a^{8}-\frac{359}{164}a^{7}+\frac{12}{41}a^{6}+\frac{107}{82}a^{5}-\frac{47}{41}a^{4}+\frac{49}{82}a^{3}+\frac{67}{41}a^{2}-\frac{8}{41}a+\frac{34}{41}$, $\frac{1}{164}a^{19}+\frac{1}{41}a^{18}-\frac{15}{164}a^{17}+\frac{35}{164}a^{16}-\frac{37}{164}a^{14}+\frac{69}{164}a^{13}-\frac{2}{41}a^{12}-\frac{1}{4}a^{11}+\frac{21}{41}a^{10}-\frac{43}{164}a^{9}-\frac{129}{164}a^{8}+\frac{5}{82}a^{7}+\frac{7}{164}a^{6}-\frac{73}{164}a^{5}+\frac{29}{82}a^{4}+\frac{49}{82}a^{3}+\frac{11}{82}a^{2}+\frac{25}{82}a-\frac{7}{41}$, $\frac{1}{82}a^{19}-\frac{33}{164}a^{18}+\frac{11}{164}a^{17}+\frac{29}{164}a^{16}-\frac{3}{4}a^{15}+\frac{2}{41}a^{14}+\frac{97}{164}a^{13}-\frac{221}{164}a^{12}+\frac{1}{2}a^{11}+\frac{291}{164}a^{10}-\frac{209}{164}a^{9}+\frac{193}{164}a^{8}+\frac{389}{164}a^{7}-\frac{58}{41}a^{6}+\frac{59}{164}a^{5}+\frac{157}{164}a^{4}-\frac{107}{82}a^{3}-\frac{19}{82}a^{2}-\frac{73}{82}a-\frac{69}{82}$, $\frac{19}{164}a^{19}-\frac{47}{164}a^{18}+\frac{43}{164}a^{17}+\frac{9}{164}a^{16}-\frac{1}{2}a^{15}+\frac{19}{41}a^{14}-\frac{1}{164}a^{13}-\frac{193}{164}a^{12}+\frac{5}{4}a^{11}-\frac{85}{164}a^{10}-\frac{79}{164}a^{9}+\frac{255}{164}a^{8}+\frac{13}{82}a^{7}-\frac{77}{82}a^{6}+\frac{171}{164}a^{5}-\frac{169}{164}a^{4}-\frac{6}{41}a^{3}+\frac{45}{82}a^{2}-\frac{17}{82}a+\frac{21}{82}$, $\frac{11}{82}a^{19}-\frac{35}{164}a^{18}+\frac{20}{41}a^{17}-\frac{25}{82}a^{16}+\frac{129}{164}a^{14}-\frac{81}{164}a^{13}-\frac{47}{82}a^{12}+a^{11}-\frac{325}{164}a^{10}-\frac{11}{41}a^{9}+\frac{8}{41}a^{8}-\frac{27}{41}a^{7}-\frac{51}{164}a^{6}+\frac{239}{164}a^{5}-\frac{9}{41}a^{4}+\frac{53}{82}a^{3}+\frac{37}{82}a^{2}+\frac{17}{82}a+\frac{51}{41}$, $\frac{1}{164}a^{19}+\frac{1}{41}a^{18}-\frac{15}{164}a^{17}+\frac{35}{164}a^{16}-\frac{39}{82}a^{14}+\frac{55}{82}a^{13}+\frac{33}{164}a^{12}-\frac{5}{4}a^{11}+\frac{83}{82}a^{10}+\frac{39}{164}a^{9}-\frac{293}{164}a^{8}+\frac{87}{82}a^{7}+\frac{65}{82}a^{6}-\frac{139}{82}a^{5}+\frac{99}{164}a^{4}+\frac{49}{82}a^{3}-\frac{71}{82}a^{2}-\frac{8}{41}a+\frac{27}{82}$, $\frac{29}{164}a^{19}-\frac{12}{41}a^{18}+\frac{4}{41}a^{17}+\frac{18}{41}a^{16}-\frac{3}{4}a^{15}+\frac{17}{82}a^{14}+\frac{37}{82}a^{13}-\frac{58}{41}a^{12}+\frac{1}{4}a^{11}+\frac{29}{82}a^{10}-\frac{35}{41}a^{9}+\frac{59}{41}a^{8}+\frac{167}{164}a^{7}+\frac{20}{41}a^{6}+\frac{69}{82}a^{5}-\frac{10}{41}a^{4}-\frac{137}{82}a^{3}+\frac{16}{41}a^{2}-\frac{68}{41}a-\frac{39}{41}$, $\frac{7}{82}a^{19}-\frac{13}{82}a^{18}+\frac{9}{41}a^{17}-\frac{1}{82}a^{16}-\frac{1}{4}a^{15}+\frac{14}{41}a^{14}+\frac{23}{164}a^{13}-\frac{97}{82}a^{12}+a^{11}-\frac{27}{82}a^{10}-\frac{137}{82}a^{9}+\frac{61}{41}a^{8}+\frac{99}{164}a^{7}-\frac{78}{41}a^{6}+\frac{413}{164}a^{5}-\frac{2}{41}a^{4}-\frac{26}{41}a^{3}+\frac{36}{41}a^{2}-\frac{19}{82}a-\frac{16}{41}$, $\frac{33}{164}a^{19}-\frac{73}{164}a^{18}+\frac{19}{82}a^{17}+\frac{89}{164}a^{16}-a^{15}+\frac{9}{164}a^{14}+\frac{227}{164}a^{13}-\frac{387}{164}a^{12}+\frac{1}{4}a^{11}+\frac{353}{164}a^{10}-\frac{78}{41}a^{9}+\frac{7}{164}a^{8}+\frac{247}{82}a^{7}-\frac{261}{164}a^{6}-\frac{195}{164}a^{5}+\frac{233}{164}a^{4}-\frac{32}{41}a^{3}-\frac{3}{41}a^{2}+\frac{5}{82}a-\frac{11}{82}$ 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:  \( 30574.1646172 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 30574.1646172 \cdot 1}{4\cdot\sqrt{2502343454824177132896256}}\cr\approx \mathstrut & 0.463361379272 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4)
 
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^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4, 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^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4);
 
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^19 + 2*x^18 + 2*x^17 - 5*x^16 + 4*x^15 + 4*x^14 - 12*x^13 + 7*x^12 + 2*x^11 - 14*x^10 + 6*x^9 + 5*x^8 - 12*x^7 + 8*x^6 + 4*x^5 - 4*x^4 + 8*x^3 + 4);
 
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

$C_2\times A_5$ (as 20T31):

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 120
The 10 conjugacy class representatives for $C_2\times A_5$
Character table for $C_2\times A_5$

Intermediate fields

\(\Q(\sqrt{-1}) \), 10.2.98867482624.1

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 10 sibling: data not computed
Degree 12 siblings: data not computed
Degree 20 sibling: data not computed
Degree 24 sibling: data not computed
Degree 30 siblings: data not computed
Degree 40 sibling: data not computed
Minimal sibling: 10.0.5473632256.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.6.0.1}{6} }^{3}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.5.0.1}{5} }^{4}$ R ${\href{/padicField/19.10.0.1}{10} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{10}$

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 2.8.14.2$x^{8} + 2 x^{7} + 2$$8$$1$$14$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.18.51$x^{12} + 2 x^{11} + 16 x^{10} + 44 x^{9} + 18 x^{8} - 8 x^{7} + 24 x^{6} + 40 x^{5} + 20 x^{4} + 8 x^{3} + 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
17.3.2.1$x^{3} + 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.2.1$x^{3} + 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 48 x^{5} + 879 x^{4} + 7682 x^{3} + 44916 x^{2} + 265428 x + 127425$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$