Properties

Label 20.0.164...016.1
Degree $20$
Signature $[0, 10]$
Discriminant $1.646\times 10^{33}$
Root discriminant \(45.79\)
Ramified primes $2,887$
Class number $32$ (GRH)
Class group [2, 4, 4] (GRH)
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 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 4)
 
gp: K = bnfinit(y^20 + 32*y^18 + 422*y^16 + 2990*y^14 + 12409*y^12 + 30722*y^10 + 43977*y^8 + 33208*y^6 + 10956*y^4 + 1216*y^2 + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 4)
 

\( x^{20} + 32 x^{18} + 422 x^{16} + 2990 x^{14} + 12409 x^{12} + 30722 x^{10} + 43977 x^{8} + 33208 x^{6} + 10956 x^{4} + 1216 x^{2} + 4 \) 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:   \(1645692992578532526654491528790016\) \(\medspace = 2^{32}\cdot 887^{8}\) 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:  \(45.79\)
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:   \(2\), \(887\) 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 a CM field.
Reflex fields:  unavailable$^{512}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{16}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{9}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{59769848}a^{18}-\frac{2136315}{29884924}a^{16}-\frac{416579}{29884924}a^{14}-\frac{1863123}{7471231}a^{12}-\frac{1}{4}a^{11}+\frac{7551227}{59769848}a^{10}-\frac{1}{2}a^{9}-\frac{2113163}{7471231}a^{8}-\frac{1}{4}a^{7}+\frac{14017109}{59769848}a^{6}-\frac{1}{4}a^{5}+\frac{2478771}{29884924}a^{4}-\frac{1}{2}a^{3}+\frac{4919591}{29884924}a^{2}+\frac{1506693}{7471231}$, $\frac{1}{59769848}a^{19}-\frac{2136315}{29884924}a^{17}-\frac{416579}{29884924}a^{15}+\frac{18739}{29884924}a^{13}-\frac{1}{4}a^{12}-\frac{7391235}{59769848}a^{11}-\frac{1}{4}a^{10}+\frac{13961041}{29884924}a^{9}+\frac{1}{4}a^{8}+\frac{14017109}{59769848}a^{7}-\frac{1}{2}a^{6}-\frac{1248115}{7471231}a^{5}-\frac{1}{4}a^{4}+\frac{4919591}{29884924}a^{3}-\frac{4457845}{14942462}a-\frac{1}{2}$ 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

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  $9$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{1920}{77023} a^{19} - \frac{233267}{308092} a^{17} - \frac{720534}{77023} a^{15} - \frac{4716681}{77023} a^{13} - \frac{35678565}{154046} a^{11} - \frac{159213449}{308092} a^{9} - \frac{102233083}{154046} a^{7} - \frac{140172083}{308092} a^{5} - \frac{22164199}{154046} a^{3} - \frac{2719329}{154046} a \)  (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:   $\frac{719273}{29884924}a^{19}+\frac{21290319}{29884924}a^{17}+\frac{126703115}{14942462}a^{15}+\frac{392417639}{7471231}a^{13}+\frac{5457011607}{29884924}a^{11}+\frac{10635887455}{29884924}a^{9}+\frac{10759229137}{29884924}a^{7}+\frac{4357530345}{29884924}a^{5}-\frac{31344978}{7471231}a^{3}-\frac{85687757}{14942462}a$, $\frac{161321}{14942462}a^{19}+\frac{9357043}{29884924}a^{17}+\frac{53862589}{14942462}a^{15}+\frac{315554261}{14942462}a^{13}+\frac{497946251}{7471231}a^{11}+\frac{3246141405}{29884924}a^{9}+\frac{554768743}{7471231}a^{7}+\frac{2892405}{29884924}a^{5}-\frac{257839613}{14942462}a^{3}-\frac{40049515}{14942462}a$, $\frac{1027251}{59769848}a^{19}+\frac{1179949}{14942462}a^{18}+\frac{7687629}{14942462}a^{17}+\frac{33941851}{14942462}a^{16}+\frac{185284433}{29884924}a^{15}+\frac{775616255}{29884924}a^{14}+\frac{580903119}{14942462}a^{13}+\frac{1131847663}{7471231}a^{12}+\frac{8158485479}{59769848}a^{11}+\frac{14411340099}{29884924}a^{10}+\frac{7996424639}{29884924}a^{9}+\frac{24554476333}{29884924}a^{8}+\frac{16297151569}{59769848}a^{7}+\frac{5042463443}{7471231}a^{6}+\frac{3546287519}{29884924}a^{5}+\frac{3132032311}{14942462}a^{4}+\frac{248608099}{29884924}a^{3}+\frac{299179283}{14942462}a^{2}-\frac{18207941}{7471231}a+\frac{1561746}{7471231}$, $\frac{2431463}{59769848}a^{19}-\frac{1474499}{29884924}a^{18}+\frac{9957515}{7471231}a^{17}-\frac{44694839}{29884924}a^{16}+\frac{269351959}{14942462}a^{15}-\frac{549275351}{29884924}a^{14}+\frac{977768083}{7471231}a^{13}-\frac{3552883771}{29884924}a^{12}+\frac{33076425931}{59769848}a^{11}-\frac{13110237381}{29884924}a^{10}+\frac{10280393552}{7471231}a^{9}-\frac{27842176559}{29884924}a^{8}+\frac{114458893131}{59769848}a^{7}-\frac{32329504765}{29884924}a^{6}+\frac{9713949176}{7471231}a^{5}-\frac{4413868439}{7471231}a^{4}+\frac{9223540599}{29884924}a^{3}-\frac{703177486}{7471231}a^{2}+\frac{326182185}{14942462}a-\frac{11384373}{7471231}$, $\frac{2754235}{59769848}a^{19}-\frac{144585}{7471231}a^{18}+\frac{39487563}{29884924}a^{17}-\frac{8524601}{14942462}a^{16}+\frac{448756465}{29884924}a^{15}-\frac{202697631}{29884924}a^{14}+\frac{1296431537}{14942462}a^{13}-\frac{315905857}{7471231}a^{12}+\frac{16131568599}{59769848}a^{11}-\frac{4493234119}{29884924}a^{10}+\frac{6435176977}{14942462}a^{9}-\frac{9245651069}{29884924}a^{8}+\frac{15994902953}{59769848}a^{7}-\frac{2625043324}{7471231}a^{6}-\frac{916919719}{14942462}a^{5}-\frac{1393428127}{7471231}a^{4}-\frac{3252740735}{29884924}a^{3}-\frac{338274915}{14942462}a^{2}-\frac{284569269}{14942462}a+\frac{7568744}{7471231}$, $\frac{2178189}{29884924}a^{18}+\frac{15691944}{7471231}a^{16}+\frac{179649919}{7471231}a^{14}+\frac{2101164629}{14942462}a^{12}+\frac{13383332583}{29884924}a^{10}+\frac{11349285657}{14942462}a^{8}+\frac{18209834641}{29884924}a^{6}+\frac{2522495027}{14942462}a^{4}+\frac{170996487}{14942462}a^{2}+\frac{1253831}{7471231}$, $\frac{2271329}{59769848}a^{19}-\frac{245051}{14942462}a^{18}+\frac{36300011}{29884924}a^{17}-\frac{3029105}{7471231}a^{16}+\frac{476774057}{29884924}a^{15}-\frac{26213548}{7471231}a^{14}+\frac{836485653}{7471231}a^{13}-\frac{310077429}{29884924}a^{12}+\frac{27229533759}{59769848}a^{11}+\frac{292788357}{14942462}a^{10}+\frac{16209558211}{14942462}a^{9}+\frac{5832117735}{29884924}a^{8}+\frac{85780390201}{59769848}a^{7}+\frac{13215299483}{29884924}a^{6}+\frac{27069710671}{29884924}a^{5}+\frac{2787479604}{7471231}a^{4}+\frac{5146944843}{29884924}a^{3}+\frac{589896037}{7471231}a^{2}-\frac{1854553}{7471231}a+\frac{1611322}{7471231}$, $\frac{2584887}{59769848}a^{19}+\frac{24920}{7471231}a^{18}+\frac{20370735}{14942462}a^{17}+\frac{1764755}{29884924}a^{16}+\frac{526926165}{29884924}a^{15}+\frac{253589}{7471231}a^{14}+\frac{3640696417}{29884924}a^{13}-\frac{171646773}{29884924}a^{12}+\frac{29241758199}{59769848}a^{11}-\frac{1506990859}{29884924}a^{10}+\frac{8655814327}{7471231}a^{9}-\frac{2831343115}{14942462}a^{8}+\frac{92864008331}{59769848}a^{7}-\frac{2596095051}{7471231}a^{6}+\frac{31319686167}{29884924}a^{5}-\frac{4255788791}{14942462}a^{4}+\frac{7903350521}{29884924}a^{3}-\frac{1080215531}{14942462}a^{2}+\frac{87622859}{7471231}a-\frac{6525875}{7471231}$, $\frac{3929335}{29884924}a^{19}+\frac{120647515}{29884924}a^{17}+\frac{755620387}{14942462}a^{15}+\frac{5030965523}{14942462}a^{13}+\frac{38844381997}{29884924}a^{11}+\frac{88803111361}{29884924}a^{9}+\frac{117315329677}{29884924}a^{7}+\frac{82891418981}{29884924}a^{5}+\frac{6624090887}{7471231}a^{3}+\frac{1430880129}{14942462}a$ 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:  \( 18267764.1868 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 18267764.1868 \cdot 32}{4\cdot\sqrt{1645692992578532526654491528790016}}\cr\approx \mathstrut & 0.345461502394 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^20 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 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 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 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 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 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 + 32*x^18 + 422*x^16 + 2990*x^14 + 12409*x^12 + 30722*x^10 + 43977*x^8 + 33208*x^6 + 10956*x^4 + 1216*x^2 + 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

\(\Q(\sqrt{-1}) \), 10.10.2535446361542656.1

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: 10.0.40567141784682496.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.5.0.1}{5} }^{4}$ ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ ${\href{/padicField/13.5.0.1}{5} }^{4}$ ${\href{/padicField/17.5.0.1}{5} }^{4}$ ${\href{/padicField/19.2.0.1}{2} }^{10}$ ${\href{/padicField/23.6.0.1}{6} }^{3}{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.5.0.1}{5} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{10}$ ${\href{/padicField/37.5.0.1}{5} }^{4}$ ${\href{/padicField/41.3.0.1}{3} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.10.0.1}{10} }^{2}$ ${\href{/padicField/47.10.0.1}{10} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.8.14.2$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}$
\(887\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$
Deg $4$$2$$2$$2$