↑

Properties

Label 20.0.330...961.1
Degree $20$
Signature $[0, 10]$
Discriminant $3.306\times 10^{22}$
Root discriminant \(13.36\)
Ramified prime $653$
Class number $1$
Class group trivial
Galois group $C_2\times A_5$ (as 20T36)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 1)
 
gp: K = bnfinit(y^20 - 2*y^19 + 4*y^18 - 6*y^17 + 10*y^16 - 15*y^15 + 21*y^14 - 27*y^13 + 37*y^12 - 34*y^11 + 47*y^10 - 34*y^9 + 37*y^8 - 27*y^7 + 21*y^6 - 15*y^5 + 10*y^4 - 6*y^3 + 4*y^2 - 2*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 2*x^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 2*x^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 1)
 

\( x^{20} - 2 x^{19} + 4 x^{18} - 6 x^{17} + 10 x^{16} - 15 x^{15} + 21 x^{14} - 27 x^{13} + 37 x^{12} + \cdots + 1 \) 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:  $[0, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(33060197995068669948961\) \(\medspace = 653^{8}\) 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:  \(13.36\)
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:  $653^{1/2}\approx 25.553864678361276$
Ramified primes:   \(653\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{137}a^{18}+\frac{14}{137}a^{17}-\frac{47}{137}a^{16}+\frac{50}{137}a^{15}+\frac{35}{137}a^{14}-\frac{53}{137}a^{13}-\frac{40}{137}a^{12}-\frac{66}{137}a^{11}-\frac{20}{137}a^{10}-\frac{14}{137}a^{9}-\frac{20}{137}a^{8}-\frac{66}{137}a^{7}-\frac{40}{137}a^{6}-\frac{53}{137}a^{5}+\frac{35}{137}a^{4}+\frac{50}{137}a^{3}-\frac{47}{137}a^{2}+\frac{14}{137}a+\frac{1}{137}$, $\frac{1}{13015}a^{19}-\frac{41}{13015}a^{18}+\frac{3978}{13015}a^{17}-\frac{4763}{13015}a^{16}-\frac{3948}{13015}a^{15}-\frac{32}{685}a^{14}+\frac{2738}{13015}a^{13}+\frac{1586}{13015}a^{12}-\frac{637}{13015}a^{11}-\frac{1791}{13015}a^{10}-\frac{3634}{13015}a^{9}+\frac{4322}{13015}a^{8}+\frac{5234}{13015}a^{7}+\frac{3517}{13015}a^{6}+\frac{112}{685}a^{5}-\frac{642}{13015}a^{4}+\frac{4053}{13015}a^{3}-\frac{278}{13015}a^{2}+\frac{3341}{13015}a-\frac{5946}{13015}$ 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:   \( -1 \)  (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:   $a$, $\frac{968}{13015}a^{19}-\frac{3493}{13015}a^{18}+\frac{10424}{13015}a^{17}-\frac{12489}{13015}a^{16}+\frac{18426}{13015}a^{15}-\frac{1291}{685}a^{14}+\frac{42254}{13015}a^{13}-\frac{55717}{13015}a^{12}+\frac{66054}{13015}a^{11}-\frac{62828}{13015}a^{10}+\frac{75283}{13015}a^{9}-\frac{28224}{13015}a^{8}+\frac{87657}{13015}a^{7}+\frac{17421}{13015}a^{6}+\frac{1971}{685}a^{5}+\frac{7634}{13015}a^{4}-\frac{6561}{13015}a^{3}+\frac{8011}{13015}a^{2}-\frac{7502}{13015}a+\frac{7067}{13015}$, $\frac{11474}{13015}a^{19}-\frac{20134}{13015}a^{18}+\frac{30937}{13015}a^{17}-\frac{41432}{13015}a^{16}+\frac{69988}{13015}a^{15}-\frac{5523}{685}a^{14}+\frac{131362}{13015}a^{13}-\frac{152626}{13015}a^{12}+\frac{220192}{13015}a^{11}-\frac{129019}{13015}a^{10}+\frac{219859}{13015}a^{9}-\frac{100262}{13015}a^{8}+\frac{75241}{13015}a^{7}-\frac{69772}{13015}a^{6}+\frac{903}{685}a^{5}-\frac{26513}{13015}a^{4}+\frac{577}{13015}a^{3}-\frac{2807}{13015}a^{2}+\frac{10399}{13015}a+\frac{8016}{13015}$, $\frac{6071}{13015}a^{19}-\frac{2291}{13015}a^{18}-\frac{1697}{13015}a^{17}+\frac{8382}{13015}a^{16}-\frac{1898}{13015}a^{15}+\frac{413}{685}a^{14}-\frac{27587}{13015}a^{13}+\frac{50136}{13015}a^{12}-\frac{35972}{13015}a^{11}+\frac{137814}{13015}a^{10}-\frac{44339}{13015}a^{9}+\frac{157087}{13015}a^{8}-\frac{80261}{13015}a^{7}+\frac{85767}{13015}a^{6}-\frac{2508}{685}a^{5}+\frac{48718}{13015}a^{4}-\frac{25837}{13015}a^{3}+\frac{22452}{13015}a^{2}-\frac{16484}{13015}a+\frac{17794}{13015}$, $\frac{7037}{13015}a^{19}-\frac{11022}{13015}a^{18}+\frac{17396}{13015}a^{17}-\frac{17856}{13015}a^{16}+\frac{31739}{13015}a^{15}-\frac{2394}{685}a^{14}+\frac{56881}{13015}a^{13}-\frac{56253}{13015}a^{12}+\frac{83131}{13015}a^{11}-\frac{10257}{13015}a^{10}+\frac{73647}{13015}a^{9}+\frac{44409}{13015}a^{8}+\frac{9658}{13015}a^{7}+\frac{48654}{13015}a^{6}-\frac{306}{685}a^{5}+\frac{27616}{13015}a^{4}-\frac{7159}{13015}a^{3}-\frac{5271}{13015}a^{2}-\frac{1028}{13015}a-\frac{7612}{13015}$, $\frac{474}{2603}a^{19}-\frac{3797}{2603}a^{18}+\frac{6472}{2603}a^{17}-\frac{9563}{2603}a^{16}+\frac{11567}{2603}a^{15}-\frac{1022}{137}a^{14}+\frac{29144}{2603}a^{13}-\frac{37705}{2603}a^{12}+\frac{43007}{2603}a^{11}-\frac{52796}{2603}a^{10}+\frac{31640}{2603}a^{9}-\frac{52373}{2603}a^{8}+\frac{17224}{2603}a^{7}-\frac{28255}{2603}a^{6}+\frac{564}{137}a^{5}-\frac{14710}{2603}a^{4}+\frac{6264}{2603}a^{3}-\frac{5118}{2603}a^{2}+\frac{1276}{2603}a+\frac{664}{2603}$, $\frac{2668}{13015}a^{19}+\frac{4232}{13015}a^{18}-\frac{17101}{13015}a^{17}+\frac{43026}{13015}a^{16}-\frac{62744}{13015}a^{15}+\frac{4734}{685}a^{14}-\frac{135511}{13015}a^{13}+\frac{207248}{13015}a^{12}-\frac{270146}{13015}a^{11}+\frac{380777}{13015}a^{10}-\frac{340577}{13015}a^{9}+\frac{434526}{13015}a^{8}-\frac{315423}{13015}a^{7}+\frac{296321}{13015}a^{6}-\frac{10589}{685}a^{5}+\frac{129384}{13015}a^{4}-\frac{112736}{13015}a^{3}+\frac{74251}{13015}a^{2}-\frac{37682}{13015}a+\frac{23872}{13015}$, $\frac{7307}{13015}a^{19}-\frac{10122}{13015}a^{18}+\frac{9596}{13015}a^{17}-\frac{5311}{13015}a^{16}+\frac{6789}{13015}a^{15}-\frac{629}{685}a^{14}+\frac{5551}{13015}a^{13}+\frac{10302}{13015}a^{12}-\frac{6874}{13015}a^{11}+\frac{112748}{13015}a^{10}-\frac{72958}{13015}a^{9}+\frac{138989}{13015}a^{8}-\frac{148082}{13015}a^{7}+\frac{63919}{13015}a^{6}-\frac{5511}{685}a^{5}+\frac{38956}{13015}a^{4}-\frac{32329}{13015}a^{3}+\frac{20844}{13015}a^{2}-\frac{11623}{13015}a+\frac{12798}{13015}$, $\frac{96}{13015}a^{19}-\frac{13816}{13015}a^{18}+\frac{22313}{13015}a^{17}-\frac{31933}{13015}a^{16}+\frac{38042}{13015}a^{15}-\frac{3462}{685}a^{14}+\frac{96693}{13015}a^{13}-\frac{116309}{13015}a^{12}+\frac{122388}{13015}a^{11}-\frac{182576}{13015}a^{10}+\frac{49756}{13015}a^{9}-\frac{155373}{13015}a^{8}-\frac{3791}{13015}a^{7}+\frac{3992}{13015}a^{6}-\frac{733}{685}a^{5}+\frac{22063}{13015}a^{4}-\frac{26822}{13015}a^{3}-\frac{4838}{13015}a^{2}+\frac{206}{13015}a-\frac{8036}{13015}$ 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:  \( 932.282807106 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 932.282807106 \cdot 1}{2\cdot\sqrt{33060197995068669948961}}\cr\approx \mathstrut & 0.245846278181 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 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^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 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^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 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^19 + 4*x^18 - 6*x^17 + 10*x^16 - 15*x^15 + 21*x^14 - 27*x^13 + 37*x^12 - 34*x^11 + 47*x^10 - 34*x^9 + 37*x^8 - 27*x^7 + 21*x^6 - 15*x^5 + 10*x^4 - 6*x^3 + 4*x^2 - 2*x + 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

$C_2\times A_5$ (as 20T36):

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

10.2.181824635281.1

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: 10.2.118731486838493.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }^{2}$ ${\href{/padicField/3.10.0.1}{10} }^{2}$ ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.2.0.1}{2} }$ ${\href{/padicField/7.3.0.1}{3} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ ${\href{/padicField/11.3.0.1}{3} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{10}$ ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.3.0.1}{3} }^{6}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.6.0.1}{6} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{2}$ ${\href{/padicField/41.6.0.1}{6} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }$ ${\href{/padicField/43.6.0.1}{6} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.5.0.1}{5} }^{4}$ ${\href{/padicField/53.5.0.1}{5} }^{4}$ ${\href{/padicField/59.5.0.1}{5} }^{4}$

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
\(653\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$