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Properties

Label 12.12.433...000.1
Degree $12$
Signature $[12, 0]$
Discriminant $4.337\times 10^{19}$
Root discriminant \(43.29\)
Ramified primes $2,3,5,101$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3^3:(C_4\times S_3)$ (as 12T170)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595)
 
gp: K = bnfinit(y^12 - 48*y^10 - 60*y^9 + 774*y^8 + 1830*y^7 - 3707*y^6 - 15300*y^5 - 6999*y^4 + 22920*y^3 + 22725*y^2 - 6060*y - 9595, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595)
 

\( x^{12} - 48 x^{10} - 60 x^{9} + 774 x^{8} + 1830 x^{7} - 3707 x^{6} - 15300 x^{5} - 6999 x^{4} + \cdots - 9595 \) Copy content Toggle raw display

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

Invariants

Degree:  $12$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[12, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(43369837128000000000\) \(\medspace = 2^{12}\cdot 3^{12}\cdot 5^{9}\cdot 101^{2}\) 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:  \(43.29\)
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:  $2\cdot 3^{25/18}5^{3/4}101^{2/3}\approx 667.0219987733675$
Ramified primes:   \(2\), \(3\), \(5\), \(101\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{945324540124}a^{11}+\frac{13792940097}{236331135031}a^{10}+\frac{157795514887}{945324540124}a^{9}-\frac{354989958125}{945324540124}a^{8}-\frac{120267445567}{472662270062}a^{7}-\frac{269081946773}{945324540124}a^{6}+\frac{29588221538}{236331135031}a^{5}+\frac{78259150946}{236331135031}a^{4}-\frac{302521257943}{945324540124}a^{3}-\frac{55011991587}{236331135031}a^{2}-\frac{33075914908}{236331135031}a+\frac{216683979963}{945324540124}$ 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:  $11$
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:   $\frac{607461120}{7623585001}a^{11}-\frac{1223111664}{7623585001}a^{10}-\frac{26716500216}{7623585001}a^{9}+\frac{17419285584}{7623585001}a^{8}+\frac{435918897630}{7623585001}a^{7}+\frac{232119728243}{7623585001}a^{6}-\frac{2731648843368}{7623585001}a^{5}-\frac{3783832268412}{7623585001}a^{4}+\frac{3446683708404}{7623585001}a^{3}+\frac{7001188748379}{7623585001}a^{2}-\frac{432234311640}{7623585001}a-\frac{2862864805482}{7623585001}$, $\frac{58902422103}{945324540124}a^{11}-\frac{87941667699}{945324540124}a^{10}-\frac{1355602571225}{472662270062}a^{9}+\frac{579333872097}{945324540124}a^{8}+\frac{45239481287073}{945324540124}a^{7}+\frac{19316456605781}{472662270062}a^{6}-\frac{70780624492464}{236331135031}a^{5}-\frac{116999980796175}{236331135031}a^{4}+\frac{329786691915951}{945324540124}a^{3}+\frac{851794376048145}{945324540124}a^{2}-\frac{4883576569563}{945324540124}a-\frac{179292022856541}{472662270062}$, $\frac{13947872994}{236331135031}a^{11}-\frac{95709626355}{945324540124}a^{10}-\frac{2508120369699}{945324540124}a^{9}+\frac{232434998001}{236331135031}a^{8}+\frac{41400895594347}{945324540124}a^{7}+\frac{31703164295713}{945324540124}a^{6}-\frac{64675335318129}{236331135031}a^{5}-\frac{103493365301817}{236331135031}a^{4}+\frac{76157302387883}{236331135031}a^{3}+\frac{759535707244977}{945324540124}a^{2}-\frac{10662485036679}{945324540124}a-\frac{318187248266183}{945324540124}$, $\frac{5242844263}{236331135031}a^{11}-\frac{65378782725}{945324540124}a^{10}-\frac{834096975865}{945324540124}a^{9}+\frac{363893945048}{236331135031}a^{8}+\frac{12871291121901}{945324540124}a^{7}-\frac{4479008755361}{945324540124}a^{6}-\frac{20307127219367}{236331135031}a^{5}-\frac{13281172578602}{236331135031}a^{4}+\frac{31685221288232}{236331135031}a^{3}+\frac{115275025136363}{945324540124}a^{2}-\frac{47668583150173}{945324540124}a-\frac{46237903755501}{945324540124}$, $\frac{82454175487}{945324540124}a^{11}-\frac{48589588767}{236331135031}a^{10}-\frac{3515551957111}{945324540124}a^{9}+\frac{3393331048013}{945324540124}a^{8}+\frac{28213699065875}{472662270062}a^{7}+\frac{16636441203245}{945324540124}a^{6}-\frac{88462223446724}{236331135031}a^{5}-\frac{105471359897149}{236331135031}a^{4}+\frac{471132805629727}{945324540124}a^{3}+\frac{200252256455855}{236331135031}a^{2}-\frac{21630422360232}{236331135031}a-\frac{322851274180379}{945324540124}$, $\frac{29494477461}{236331135031}a^{11}-\frac{97813196045}{945324540124}a^{10}-\frac{5709569317553}{945324540124}a^{9}-\frac{465770088805}{236331135031}a^{8}+\frac{97620643262289}{945324540124}a^{7}+\frac{122963008030219}{945324540124}a^{6}-\frac{152390343807548}{236331135031}a^{5}-\frac{306455664126340}{236331135031}a^{4}+\frac{157896598636308}{236331135031}a^{3}+\frac{22\!\cdots\!15}{945324540124}a^{2}+\frac{98551810525971}{945324540124}a-\frac{951507357178429}{945324540124}$, $\frac{79513963603}{472662270062}a^{11}-\frac{444076946169}{945324540124}a^{10}-\frac{6485183633327}{945324540124}a^{9}+\frac{4464569837625}{472662270062}a^{8}+\frac{101475179491569}{945324540124}a^{7}-\frac{1247858835287}{945324540124}a^{6}-\frac{158493361257231}{236331135031}a^{5}-\frac{152036047777812}{236331135031}a^{4}+\frac{440789287770121}{472662270062}a^{3}+\frac{11\!\cdots\!71}{945324540124}a^{2}-\frac{213864083831697}{945324540124}a-\frac{473548562446507}{945324540124}$, $\frac{37589478201}{945324540124}a^{11}-\frac{64338694531}{472662270062}a^{10}-\frac{1427712857451}{945324540124}a^{9}+\frac{2902079230319}{945324540124}a^{8}+\frac{5346165913583}{236331135031}a^{7}-\frac{11175267963783}{945324540124}a^{6}-\frac{33139348796425}{236331135031}a^{5}-\frac{19098185446165}{236331135031}a^{4}+\frac{195343492539137}{945324540124}a^{3}+\frac{82921826209463}{472662270062}a^{2}-\frac{31827229171469}{472662270062}a-\frac{58974525149031}{945324540124}$, $\frac{690965043723}{945324540124}a^{11}-\frac{758047832193}{472662270062}a^{10}-\frac{29832004414193}{945324540124}a^{9}+\frac{23963070267937}{945324540124}a^{8}+\frac{120491386133147}{236331135031}a^{7}+\frac{207888535047127}{945324540124}a^{6}-\frac{753479761122148}{236331135031}a^{5}-\frac{991483828816775}{236331135031}a^{4}+\frac{38\!\cdots\!11}{945324540124}a^{3}+\frac{37\!\cdots\!15}{472662270062}a^{2}-\frac{265599558142391}{472662270062}a-\frac{30\!\cdots\!53}{945324540124}$, $\frac{129819775275}{236331135031}a^{11}-\frac{976779668753}{945324540124}a^{10}-\frac{23090365112441}{945324540124}a^{9}+\frac{3071655309744}{236331135031}a^{8}+\frac{378899234541205}{945324540124}a^{7}+\frac{237315517854975}{945324540124}a^{6}-\frac{593192961826458}{236331135031}a^{5}-\frac{870091589593551}{236331135031}a^{4}+\frac{730430258960758}{236331135031}a^{3}+\frac{64\!\cdots\!63}{945324540124}a^{2}-\frac{270408452104193}{945324540124}a-\frac{26\!\cdots\!09}{945324540124}$, $\frac{402243435619}{472662270062}a^{11}-\frac{913482871965}{472662270062}a^{10}-\frac{8623434293599}{236331135031}a^{9}+\frac{15080665792223}{472662270062}a^{8}+\frac{277603081948347}{472662270062}a^{7}+\frac{52276982885692}{236331135031}a^{6}-\frac{868055899375182}{236331135031}a^{5}-\frac{11\!\cdots\!00}{236331135031}a^{4}+\frac{22\!\cdots\!99}{472662270062}a^{3}+\frac{41\!\cdots\!97}{472662270062}a^{2}-\frac{337957862056715}{472662270062}a-\frac{844408195734123}{236331135031}$ 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:  \( 1110052.15261 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1110052.15261 \cdot 1}{2\cdot\sqrt{43369837128000000000}}\cr\approx \mathstrut & 0.345206877932 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595)
 
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^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595, 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^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595);
 
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^12 - 48*x^10 - 60*x^9 + 774*x^8 + 1830*x^7 - 3707*x^6 - 15300*x^5 - 6999*x^4 + 22920*x^3 + 22725*x^2 - 6060*x - 9595);
 
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

$C_3^3:(C_4\times S_3)$ (as 12T170):

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 648
The 30 conjugacy class representatives for $C_3^3:(C_4\times S_3)$
Character table for $C_3^3:(C_4\times S_3)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\)

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 12 siblings: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 36 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 R R ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ ${\href{/padicField/43.12.0.1}{12} }$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.4.0.1}{4} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 2.4.4.2$x^{4} + 4 x^{3} + 4 x^{2} + 12$$2$$2$$4$$C_4$$[2]^{2}$
2.8.8.1$x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
\(3\) Copy content Toggle raw display 3.12.12.9$x^{12} + 6 x^{11} - 12 x^{10} - 60 x^{9} + 261 x^{8} + 540 x^{7} + 540 x^{6} + 1728 x^{5} + 2187 x^{4} - 1674 x^{3} + 1296 x^{2} - 324 x + 81$$3$$4$$12$12T41$[3/2, 3/2]_{2}^{4}$
\(5\) Copy content Toggle raw display 5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
\(101\) Copy content Toggle raw display $\Q_{101}$$x + 99$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 99$$1$$1$$0$Trivial$[\ ]$
$\Q_{101}$$x + 99$$1$$1$$0$Trivial$[\ ]$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.2.0.1$x^{2} + 97 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
101.3.2.1$x^{3} + 101$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$