Properties

Label 20.6.131...624.1
Degree $20$
Signature $[6, 7]$
Discriminant $-1.317\times 10^{25}$
Root discriminant \(18.03\)
Ramified primes $2,11,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^5.C_2^8:C_{10}$ (as 20T749)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*x^2 - 1)
 
gp: K = bnfinit(y^20 + y^18 + 3*y^16 + 7*y^14 - 3*y^12 - 4*y^10 - 13*y^8 - 25*y^6 + 7*y^4 + 4*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*x^2 - 1)
 

\( x^{20} + x^{18} + 3x^{16} + 7x^{14} - 3x^{12} - 4x^{10} - 13x^{8} - 25x^{6} + 7x^{4} + 4x^{2} - 1 \) 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:  $[6, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-13167225195265942636954624\) \(\medspace = -\,2^{10}\cdot 11^{16}\cdot 23^{4}\) 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:  \(18.03\)
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:  $2^{31/16}11^{4/5}23^{1/2}\approx 125.0903140152616$
Ramified primes:   \(2\), \(11\), \(23\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{5722}a^{18}-\frac{275}{2861}a^{16}-\frac{213}{5722}a^{14}-\frac{1}{2}a^{13}+\frac{69}{5722}a^{12}-\frac{1}{2}a^{11}-\frac{829}{5722}a^{10}-\frac{1}{2}a^{9}-\frac{985}{5722}a^{8}-\frac{434}{2861}a^{6}+\frac{228}{2861}a^{4}-\frac{1171}{2861}a^{2}-\frac{1}{2}a-\frac{1363}{2861}$, $\frac{1}{5722}a^{19}-\frac{275}{2861}a^{17}-\frac{213}{5722}a^{15}-\frac{1}{2}a^{14}+\frac{69}{5722}a^{13}-\frac{1}{2}a^{12}-\frac{829}{5722}a^{11}-\frac{1}{2}a^{10}-\frac{985}{5722}a^{9}-\frac{434}{2861}a^{7}+\frac{228}{2861}a^{5}-\frac{1171}{2861}a^{3}-\frac{1}{2}a^{2}-\frac{1363}{2861}a$ 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 $1$ (assuming GRH)

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:  $12$
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:   $a^{19}+a^{17}+3a^{15}+7a^{13}-3a^{11}-4a^{9}-13a^{7}-25a^{5}+7a^{3}+4a$, $\frac{6945}{2861}a^{19}+\frac{8268}{2861}a^{17}+\frac{22740}{2861}a^{15}+\frac{52916}{2861}a^{13}-\frac{9656}{2861}a^{11}-\frac{28784}{2861}a^{9}-\frac{97407}{2861}a^{7}-\frac{191894}{2861}a^{5}+\frac{5317}{2861}a^{3}+\frac{27777}{2861}a$, $\frac{3372}{2861}a^{18}+\frac{5050}{2861}a^{16}+\frac{11319}{2861}a^{14}+\frac{29537}{2861}a^{12}-\frac{191}{2861}a^{10}-\frac{16965}{2861}a^{8}-\frac{45869}{2861}a^{6}-\frac{110304}{2861}a^{4}-\frac{9447}{2861}a^{2}+\frac{14626}{2861}$, $\frac{384}{2861}a^{19}+\frac{514}{2861}a^{17}+\frac{1177}{2861}a^{15}+\frac{3608}{2861}a^{13}-\frac{765}{2861}a^{11}-\frac{588}{2861}a^{9}-\frac{4297}{2861}a^{7}-\frac{13722}{2861}a^{5}+\frac{1887}{2861}a^{3}-\frac{5380}{2861}a$, $\frac{4084}{2861}a^{19}+\frac{5407}{2861}a^{17}+\frac{14157}{2861}a^{15}+\frac{32889}{2861}a^{13}-\frac{1073}{2861}a^{11}-\frac{17340}{2861}a^{9}-\frac{60214}{2861}a^{7}-\frac{120369}{2861}a^{5}-\frac{14710}{2861}a^{3}+\frac{16333}{2861}a$, $\frac{3245}{5722}a^{19}+\frac{1871}{2861}a^{18}+\frac{3375}{5722}a^{17}+\frac{4681}{5722}a^{16}+\frac{4880}{2861}a^{15}+\frac{12617}{5722}a^{14}+\frac{23635}{5722}a^{13}+\frac{14659}{2861}a^{12}-\frac{4674}{2861}a^{11}-\frac{3655}{5722}a^{10}-\frac{6016}{2861}a^{9}-\frac{15207}{5722}a^{8}-\frac{20745}{2861}a^{7}-\frac{27590}{2861}a^{6}-\frac{85247}{5722}a^{5}-\frac{110383}{5722}a^{4}+\frac{10957}{2861}a^{3}-\frac{1691}{2861}a^{2}+\frac{8925}{5722}a+\frac{10217}{5722}$, $\frac{2900}{2861}a^{18}+\frac{4299}{2861}a^{16}+\frac{8859}{2861}a^{14}+\frac{25579}{2861}a^{12}-\frac{3721}{2861}a^{10}-\frac{15527}{2861}a^{8}-\frac{36713}{2861}a^{6}-\frac{96656}{2861}a^{4}+\frac{8797}{2861}a^{2}+\frac{8126}{2861}$, $\frac{13049}{5722}a^{19}-\frac{2049}{5722}a^{18}+\frac{7802}{2861}a^{17}-\frac{3145}{5722}a^{16}+\frac{22185}{2861}a^{15}-\frac{3509}{2861}a^{14}+\frac{99301}{5722}a^{13}-\frac{9179}{2861}a^{12}-\frac{5812}{2861}a^{11}-\frac{813}{5722}a^{10}-\frac{25145}{2861}a^{9}+\frac{9843}{5722}a^{8}-\frac{188659}{5722}a^{7}+\frac{13800}{2861}a^{6}-\frac{358161}{5722}a^{5}+\frac{69867}{5722}a^{4}-\frac{8361}{2861}a^{3}+\frac{4722}{2861}a^{2}+\frac{23938}{2861}a-\frac{2410}{2861}$, $\frac{15007}{5722}a^{19}+\frac{1488}{2861}a^{18}+\frac{10081}{2861}a^{17}+\frac{2553}{5722}a^{16}+\frac{25372}{2861}a^{15}+\frac{9837}{5722}a^{14}+\frac{61412}{2861}a^{13}+\frac{19379}{5722}a^{12}-\frac{6897}{5722}a^{11}-\frac{3322}{2861}a^{10}-\frac{64911}{5722}a^{9}-\frac{3709}{2861}a^{8}-\frac{211657}{5722}a^{7}-\frac{21300}{2861}a^{6}-\frac{449497}{5722}a^{5}-\frac{64861}{5722}a^{4}-\frac{15240}{2861}a^{3}+\frac{2663}{2861}a^{2}+\frac{51855}{5722}a+\frac{3471}{2861}$, $\frac{7043}{2861}a^{19}+\frac{875}{2861}a^{18}+\frac{8727}{2861}a^{17}+\frac{1657}{5722}a^{16}+\frac{46647}{5722}a^{15}+\frac{2451}{2861}a^{14}+\frac{110773}{5722}a^{13}+\frac{6016}{2861}a^{12}-\frac{7929}{2861}a^{11}-\frac{5945}{5722}a^{10}-\frac{28040}{2861}a^{9}-\frac{3575}{2861}a^{8}-\frac{196143}{5722}a^{7}-\frac{9918}{2861}a^{6}-\frac{405991}{5722}a^{5}-\frac{40273}{5722}a^{4}+\frac{1820}{2861}a^{3}+\frac{12757}{5722}a^{2}+\frac{44821}{5722}a+\frac{824}{2861}$, $\frac{3245}{5722}a^{19}+\frac{548}{2861}a^{18}+\frac{3375}{5722}a^{17}+\frac{871}{5722}a^{16}+\frac{4880}{2861}a^{15}+\frac{4015}{5722}a^{14}+\frac{23635}{5722}a^{13}+\frac{3480}{2861}a^{12}-\frac{4674}{2861}a^{11}-\frac{1647}{5722}a^{10}-\frac{6016}{2861}a^{9}-\frac{963}{5722}a^{8}-\frac{20745}{2861}a^{7}-\frac{9321}{2861}a^{6}-\frac{85247}{5722}a^{5}-\frac{23787}{5722}a^{4}+\frac{10957}{2861}a^{3}-\frac{1688}{2861}a^{2}+\frac{8925}{5722}a-\frac{3673}{5722}$, $\frac{3409}{2861}a^{19}+\frac{65}{2861}a^{18}+\frac{6593}{5722}a^{17}+\frac{25}{5722}a^{16}+\frac{21181}{5722}a^{15}+\frac{460}{2861}a^{14}+\frac{23507}{2861}a^{13}+\frac{387}{5722}a^{12}-\frac{18813}{5722}a^{11}+\frac{474}{2861}a^{10}-\frac{23851}{5722}a^{9}+\frac{695}{5722}a^{8}-\frac{46514}{2861}a^{7}-\frac{2061}{2861}a^{6}-\frac{166837}{5722}a^{5}-\frac{801}{5722}a^{4}+\frac{18339}{2861}a^{3}-\frac{4055}{5722}a^{2}+\frac{19215}{5722}a+\frac{3245}{5722}$ Copy content Toggle raw display (assuming GRH)
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:  \( 59115.4547078 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{7}\cdot 59115.4547078 \cdot 1}{2\cdot\sqrt{13167225195265942636954624}}\cr\approx \mathstrut & 0.201540791436 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*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 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*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 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*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 + x^18 + 3*x^16 + 7*x^14 - 3*x^12 - 4*x^10 - 13*x^8 - 25*x^6 + 7*x^4 + 4*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

$C_2^5.C_2^8:C_{10}$ (as 20T749):

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 solvable group of order 81920
The 332 conjugacy class representatives for $C_2^5.C_2^8:C_{10}$
Character table for $C_2^5.C_2^8:C_{10}$

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.6.113395848049.1

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 20 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{2}$ ${\href{/padicField/5.10.0.1}{10} }^{2}$ ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.5.0.1}{5} }^{2}$ R ${\href{/padicField/13.10.0.1}{10} }^{2}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }{,}\,{\href{/padicField/19.5.0.1}{5} }^{2}$ R ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$

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 2.10.10.13$x^{10} + 10 x^{9} + 10 x^{8} + 56 x^{7} + 192 x^{6} + 800 x^{5} + 1536 x^{4} + 2208 x^{3} + 2224 x^{2} + 96 x - 1056$$2$$5$$10$$C_2 \times (C_2^4 : C_5)$$[2, 2, 2, 2, 2]^{5}$
2.10.0.1$x^{10} + x^{6} + x^{5} + x^{3} + x^{2} + x + 1$$1$$10$$0$$C_{10}$$[\ ]^{10}$
\(11\) Copy content Toggle raw display 11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} + 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.8.5$x^{10} + 35 x^{9} + 500 x^{8} + 3710 x^{7} + 14985 x^{6} + 31389 x^{5} + 30355 x^{4} + 19790 x^{3} + 37110 x^{2} + 111495 x + 148840$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$
23.4.2.1$x^{4} + 42 x^{3} + 497 x^{2} + 1176 x + 10467$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.0.1$x^{4} + 3 x^{2} + 19 x + 5$$1$$4$$0$$C_4$$[\ ]^{4}$