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Properties

Label 12.12.967...337.1
Degree $12$
Signature $[12, 0]$
Discriminant $9.674\times 10^{19}$
Root discriminant \(46.29\)
Ramified primes $13,17$
Class number $1$
Class group trivial
Galois group $C_{12}$ (as 12T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13)
 
gp: K = bnfinit(y^12 - y^11 - 36*y^10 + 27*y^9 + 394*y^8 - 223*y^7 - 1578*y^6 + 448*y^5 + 2307*y^4 + 161*y^3 - 806*y^2 - 130*y + 13, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13)
 

\( x^{12} - x^{11} - 36 x^{10} + 27 x^{9} + 394 x^{8} - 223 x^{7} - 1578 x^{6} + 448 x^{5} + 2307 x^{4} + \cdots + 13 \) 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:   \(96735773996756764337\) \(\medspace = 13^{8}\cdot 17^{9}\) 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:  \(46.29\)
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:  $13^{2/3}17^{3/4}\approx 46.28769904177523$
Ramified primes:   \(13\), \(17\) 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:  \(\Q(\sqrt{17}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(35,·)$, $\chi_{221}(55,·)$, $\chi_{221}(16,·)$, $\chi_{221}(81,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$, $\chi_{221}(152,·)$, $\chi_{221}(217,·)$, $\chi_{221}(157,·)$, $\chi_{221}(120,·)$, $\chi_{221}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}$, $\frac{1}{75866712768914}a^{11}-\frac{1942026629465}{37933356384457}a^{10}-\frac{7870598773781}{37933356384457}a^{9}+\frac{10063355884973}{37933356384457}a^{8}-\frac{11249884673311}{75866712768914}a^{7}-\frac{8394725357827}{75866712768914}a^{6}-\frac{18792187102377}{37933356384457}a^{5}-\frac{127091366651}{37933356384457}a^{4}-\frac{16826114168744}{37933356384457}a^{3}+\frac{12019764729589}{37933356384457}a^{2}-\frac{5893976261595}{75866712768914}a+\frac{10248914708540}{37933356384457}$ 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:  $11$
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:   $\frac{510353099531}{37933356384457}a^{11}-\frac{585867876369}{37933356384457}a^{10}-\frac{18298175170166}{37933356384457}a^{9}+\frac{16664494610623}{37933356384457}a^{8}+\frac{199089393746483}{37933356384457}a^{7}-\frac{149263732165508}{37933356384457}a^{6}-\frac{790386870014119}{37933356384457}a^{5}+\frac{401689365773603}{37933356384457}a^{4}+\frac{11\!\cdots\!68}{37933356384457}a^{3}-\frac{230881727005371}{37933356384457}a^{2}-\frac{436650027774856}{37933356384457}a+\frac{21018609717712}{37933356384457}$, $\frac{695220930595}{37933356384457}a^{11}-\frac{947349571163}{37933356384457}a^{10}-\frac{24379735126424}{37933356384457}a^{9}+\frac{27278753347726}{37933356384457}a^{8}+\frac{253328936247653}{37933356384457}a^{7}-\frac{237595694324472}{37933356384457}a^{6}-\frac{900628392996763}{37933356384457}a^{5}+\frac{556970262829275}{37933356384457}a^{4}+\frac{10\!\cdots\!14}{37933356384457}a^{3}-\frac{69696292447352}{37933356384457}a^{2}-\frac{184948842805204}{37933356384457}a-\frac{39270743397734}{37933356384457}$, $\frac{34803738330}{37933356384457}a^{11}-\frac{571778152696}{37933356384457}a^{10}-\frac{319343500110}{37933356384457}a^{9}+\frac{19287745186240}{37933356384457}a^{8}-\frac{14342965774450}{37933356384457}a^{7}-\frac{188897948298568}{37933356384457}a^{6}+\frac{200541877067258}{37933356384457}a^{5}+\frac{588282580650355}{37933356384457}a^{4}-\frac{599746880441960}{37933356384457}a^{3}-\frac{521647058349391}{37933356384457}a^{2}+\frac{321163538335367}{37933356384457}a+\frac{90426547098989}{37933356384457}$, $\frac{811446745941}{75866712768914}a^{11}-\frac{524644244367}{75866712768914}a^{10}-\frac{14915156736443}{37933356384457}a^{9}+\frac{5985231549474}{37933356384457}a^{8}+\frac{338581883962293}{75866712768914}a^{7}-\frac{39299194058751}{37933356384457}a^{6}-\frac{14\!\cdots\!51}{75866712768914}a^{5}+\frac{2005559932209}{37933356384457}a^{4}+\frac{11\!\cdots\!49}{37933356384457}a^{3}+\frac{270247490696118}{37933356384457}a^{2}-\frac{604974693685141}{75866712768914}a-\frac{116428109908917}{75866712768914}$, $\frac{1971367898323}{75866712768914}a^{11}-\frac{3983492607889}{75866712768914}a^{10}-\frac{33852018906829}{37933356384457}a^{9}+\frac{61225216532577}{37933356384457}a^{8}+\frac{679388808357459}{75866712768914}a^{7}-\frac{566358822821395}{37933356384457}a^{6}-\frac{22\!\cdots\!57}{75866712768914}a^{5}+\frac{15\!\cdots\!22}{37933356384457}a^{4}+\frac{10\!\cdots\!97}{37933356384457}a^{3}-\frac{10\!\cdots\!35}{37933356384457}a^{2}-\frac{269487308662299}{75866712768914}a+\frac{59715159615721}{75866712768914}$, $\frac{743971678535}{37933356384457}a^{11}-\frac{1556520943221}{37933356384457}a^{10}-\frac{24717553038479}{37933356384457}a^{9}+\frac{48020555703591}{37933356384457}a^{8}+\frac{227848354747199}{37933356384457}a^{7}-\frac{449651113039179}{37933356384457}a^{6}-\frac{559290488687624}{37933356384457}a^{5}+\frac{12\!\cdots\!07}{37933356384457}a^{4}-\frac{47710779397987}{37933356384457}a^{3}-\frac{898055992920298}{37933356384457}a^{2}+\frac{460625547404954}{37933356384457}a-\frac{36948952464217}{37933356384457}$, $\frac{62840471613}{37933356384457}a^{11}+\frac{235793571089}{37933356384457}a^{10}-\frac{2853566254067}{37933356384457}a^{9}-\frac{7947359427491}{37933356384457}a^{8}+\frac{42270054339596}{37933356384457}a^{7}+\frac{68090652524552}{37933356384457}a^{6}-\frac{249951514077438}{37933356384457}a^{5}-\frac{113887699056641}{37933356384457}a^{4}+\frac{472856543219368}{37933356384457}a^{3}-\frac{163073588146709}{37933356384457}a^{2}-\frac{66719299090461}{37933356384457}a+\frac{14962708016767}{37933356384457}$, $\frac{137585486205}{75866712768914}a^{11}-\frac{208736264404}{37933356384457}a^{10}-\frac{3973915430231}{75866712768914}a^{9}+\frac{13112555934025}{75866712768914}a^{8}+\frac{11097517784025}{37933356384457}a^{7}-\frac{61377578145168}{37933356384457}a^{6}+\frac{48188504450488}{37933356384457}a^{5}+\frac{340490877325427}{75866712768914}a^{4}-\frac{657650337685131}{75866712768914}a^{3}-\frac{114605098672585}{75866712768914}a^{2}+\frac{187259287823772}{37933356384457}a+\frac{675090121791}{75866712768914}$, $\frac{761236507291}{75866712768914}a^{11}-\frac{287830634730}{37933356384457}a^{10}-\frac{13102841275750}{37933356384457}a^{9}+\frac{7044900236869}{37933356384457}a^{8}+\frac{257599024218635}{75866712768914}a^{7}-\frac{112899077379949}{75866712768914}a^{6}-\frac{396165741073800}{37933356384457}a^{5}+\frac{108296048616395}{37933356384457}a^{4}+\frac{278063283201535}{37933356384457}a^{3}-\frac{7725681698793}{37933356384457}a^{2}+\frac{41672564265587}{75866712768914}a-\frac{3561075620431}{37933356384457}$, $\frac{978259871159}{75866712768914}a^{11}-\frac{298577075561}{37933356384457}a^{10}-\frac{17912845456947}{37933356384457}a^{9}+\frac{6423908725579}{37933356384457}a^{8}+\frac{403985373583763}{75866712768914}a^{7}-\frac{72659653919963}{75866712768914}a^{6}-\frac{859132571762306}{37933356384457}a^{5}-\frac{56405111984405}{37933356384457}a^{4}+\frac{13\!\cdots\!51}{37933356384457}a^{3}+\frac{422621537488524}{37933356384457}a^{2}-\frac{10\!\cdots\!05}{75866712768914}a-\frac{141356228669640}{37933356384457}$, $\frac{1927558989619}{37933356384457}a^{11}-\frac{1945932059587}{37933356384457}a^{10}-\frac{69604788434245}{37933356384457}a^{9}+\frac{52469926418941}{37933356384457}a^{8}+\frac{766682838712338}{37933356384457}a^{7}-\frac{427350023145544}{37933356384457}a^{6}-\frac{31\!\cdots\!99}{37933356384457}a^{5}+\frac{798208708528270}{37933356384457}a^{4}+\frac{45\!\cdots\!59}{37933356384457}a^{3}+\frac{521324174510198}{37933356384457}a^{2}-\frac{15\!\cdots\!21}{37933356384457}a-\frac{356676928793026}{37933356384457}$ 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:  \( 1140840.44185 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 1140840.44185 \cdot 1}{2\cdot\sqrt{96735773996756764337}}\cr\approx \mathstrut & 0.237553429737 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13)
 
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 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13, 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 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13);
 
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 - x^11 - 36*x^10 + 27*x^9 + 394*x^8 - 223*x^7 - 1578*x^6 + 448*x^5 + 2307*x^4 + 161*x^3 - 806*x^2 - 130*x + 13);
 
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

$C_{12}$ (as 12T1):

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 cyclic group of order 12
The 12 conjugacy class representatives for $C_{12}$
Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.4913.1, 6.6.140320193.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.12.0.1}{12} }$ ${\href{/padicField/5.4.0.1}{4} }^{3}$ ${\href{/padicField/7.12.0.1}{12} }$ ${\href{/padicField/11.12.0.1}{12} }$ R R ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.12.0.1}{12} }$ ${\href{/padicField/31.4.0.1}{4} }^{3}$ ${\href{/padicField/37.12.0.1}{12} }$ ${\href{/padicField/41.12.0.1}{12} }$ ${\href{/padicField/43.6.0.1}{6} }^{2}$ ${\href{/padicField/47.1.0.1}{1} }^{12}$ ${\href{/padicField/53.2.0.1}{2} }^{6}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(13\) Copy content Toggle raw display 13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} + 13$$3$$1$$2$$C_3$$[\ ]_{3}$
\(17\) Copy content Toggle raw display 17.12.9.1$x^{12} + 4 x^{10} + 56 x^{9} + 57 x^{8} + 168 x^{7} + 1044 x^{6} - 11256 x^{5} + 3356 x^{4} + 10080 x^{3} + 97736 x^{2} + 58576 x + 57252$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$