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Properties

Label 12.0.395...125.1
Degree $12$
Signature $[0, 6]$
Discriminant $3.955\times 10^{25}$
Root discriminant \(135.86\)
Ramified primes $5,7,37$
Class number $1443$ (GRH)
Class group [1443] (GRH)
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 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416)
 
gp: K = bnfinit(y^12 - y^11 + 87*y^10 - 221*y^9 + 7751*y^8 + 27358*y^7 + 649836*y^6 + 1330904*y^5 + 53241808*y^4 + 30023808*y^3 + 16929792*y^2 + 9510912*y + 5308416, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - x^11 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416)
 

\( x^{12} - x^{11} + 87 x^{10} - 221 x^{9} + 7751 x^{8} + 27358 x^{7} + 649836 x^{6} + 1330904 x^{5} + \cdots + 5308416 \) 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:  $[0, 6]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(39548330211803192814453125\) \(\medspace = 5^{9}\cdot 7^{8}\cdot 37^{8}\) 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:  \(135.86\)
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:  $5^{3/4}7^{2/3}37^{2/3}\approx 135.86075153822173$
Ramified primes:   \(5\), \(7\), \(37\) 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:  \(\Q(\sqrt{5}) \)
$\card{ \Gal(K/\Q) }$:  $12$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1295=5\cdot 7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{1295}(1,·)$, $\chi_{1295}(676,·)$, $\chi_{1295}(359,·)$, $\chi_{1295}(618,·)$, $\chi_{1295}(519,·)$, $\chi_{1295}(1037,·)$, $\chi_{1295}(877,·)$, $\chi_{1295}(1136,·)$, $\chi_{1295}(417,·)$, $\chi_{1295}(1194,·)$, $\chi_{1295}(778,·)$, $\chi_{1295}(158,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\zeta_{5})\)$^{2}$, 12.0.39548330211803192814453125.1$^{30}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{6}a^{5}+\frac{1}{6}a^{4}+\frac{1}{6}a^{3}+\frac{1}{6}a^{2}+\frac{1}{6}a$, $\frac{1}{12}a^{6}-\frac{1}{12}a^{5}-\frac{1}{12}a^{4}-\frac{1}{12}a^{3}-\frac{1}{12}a^{2}-\frac{1}{6}a$, $\frac{1}{24}a^{7}-\frac{1}{24}a^{6}-\frac{1}{24}a^{5}+\frac{11}{24}a^{4}-\frac{1}{24}a^{3}-\frac{1}{12}a^{2}-\frac{1}{2}a$, $\frac{1}{48}a^{8}-\frac{1}{48}a^{7}-\frac{1}{48}a^{6}+\frac{1}{16}a^{5}-\frac{3}{16}a^{4}+\frac{7}{24}a^{3}-\frac{5}{12}a^{2}+\frac{1}{3}a$, $\frac{1}{10063500369024}a^{9}-\frac{11463106107}{3354500123008}a^{8}-\frac{49054366947}{3354500123008}a^{7}-\frac{98147727519}{3354500123008}a^{6}-\frac{215798613779}{3354500123008}a^{5}+\frac{2162139328879}{5031750184512}a^{4}-\frac{43283118819}{838625030752}a^{3}+\frac{155133101577}{419312515376}a^{2}+\frac{66216372311}{209656257688}a-\frac{551042508}{26207032211}$, $\frac{1}{241524008856576}a^{10}-\frac{1}{241524008856576}a^{9}+\frac{588017814749}{80508002952192}a^{8}+\frac{4287874190179}{241524008856576}a^{7}+\frac{6531716848775}{241524008856576}a^{6}-\frac{2886722667505}{120762004428288}a^{5}+\frac{1101420038675}{6709000246016}a^{4}+\frac{8018758954267}{30190501107072}a^{3}-\frac{769943812691}{15095250553536}a^{2}-\frac{253967973517}{628968773064}a+\frac{1470787444}{26207032211}$, $\frac{1}{57\!\cdots\!24}a^{11}-\frac{1}{57\!\cdots\!24}a^{10}+\frac{29}{19\!\cdots\!08}a^{9}-\frac{17720222796509}{57\!\cdots\!24}a^{8}-\frac{98709841360825}{57\!\cdots\!24}a^{7}+\frac{91692203357807}{28\!\cdots\!12}a^{6}-\frac{2585882947325}{161016005904384}a^{5}+\frac{160037805055003}{724572026569728}a^{4}+\frac{70876502832445}{362286013284864}a^{3}-\frac{3148122353813}{15095250553536}a^{2}-\frac{122923678075}{628968773064}a-\frac{12155112948}{26207032211}$ 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

$C_{1443}$, which has order $1443$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $481$ (assuming GRH)

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{338597}{483048017713152} a^{11} - \frac{9819313}{161016005904384} a^{10} + \frac{74829937}{483048017713152} a^{9} - \frac{2624465347}{483048017713152} a^{8} + \frac{1162288961}{53672001968128} a^{7} - \frac{6112014447}{13418000492032} a^{6} - \frac{56330012711}{60381002214144} a^{5} - \frac{1126719778961}{30190501107072} a^{4} - \frac{26473883639}{1257937546128} a^{3} - \frac{47647220049443}{15095250553536} a^{2} - \frac{174716052}{26207032211} a - \frac{97515936}{26207032211} \)  (order $10$) 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:   $\frac{622066337}{57\!\cdots\!24}a^{11}-\frac{969266153}{57\!\cdots\!24}a^{10}+\frac{18155657045}{19\!\cdots\!08}a^{9}-\frac{167578949749}{57\!\cdots\!24}a^{8}+\frac{4898367337423}{57\!\cdots\!24}a^{7}+\frac{7163672536915}{28\!\cdots\!12}a^{6}+\frac{32895200633717}{483048017713152}a^{5}+\frac{75285954568559}{724572026569728}a^{4}+\frac{20\!\cdots\!73}{362286013284864}a^{3}+\frac{41504844671}{1257937546128}a^{2}+\frac{969266153}{52414064422}a+\frac{26380632119}{26207032211}$, $\frac{21171697}{241524008856576}a^{11}-\frac{941793841}{80508002952192}a^{10}+\frac{7203688817}{241524008856576}a^{9}-\frac{252650642627}{241524008856576}a^{8}+\frac{335988883651}{80508002952192}a^{7}-\frac{910216514245}{10063500369024}a^{6}-\frac{5350300231351}{30190501107072}a^{5}-\frac{108466464051601}{15095250553536}a^{4}-\frac{2548573834999}{628968773064}a^{3}-\frac{45\!\cdots\!35}{7547625276768}a^{2}+\frac{43790163193399}{209656257688}a-\frac{5044086947750}{26207032211}$, $\frac{4440978847}{28\!\cdots\!12}a^{11}-\frac{2948193883}{28\!\cdots\!12}a^{10}+\frac{128308852919}{966096035426304}a^{9}-\frac{859258476167}{28\!\cdots\!12}a^{8}+\frac{34132057041413}{28\!\cdots\!12}a^{7}+\frac{66242947136111}{14\!\cdots\!56}a^{6}+\frac{27106039899313}{26836000984064}a^{5}+\frac{859009512857155}{362286013284864}a^{4}+\frac{14\!\cdots\!45}{181143006642432}a^{3}+\frac{11\!\cdots\!97}{15095250553536}a^{2}+\frac{8204636658385}{314484386532}a+\frac{4774501032}{26207032211}$, $\frac{109660237}{19\!\cdots\!08}a^{11}+\frac{18184537}{644064023617536}a^{10}+\frac{8293114067}{19\!\cdots\!08}a^{9}-\frac{9104156561}{19\!\cdots\!08}a^{8}+\frac{91976908811}{214688007872512}a^{7}+\frac{629690331229}{322032011808768}a^{6}+\frac{17812231707379}{483048017713152}a^{5}+\frac{32680104087235}{241524008856576}a^{4}+\frac{118579256054335}{40254001476096}a^{3}+\frac{16868968635635}{3773812638384}a^{2}+\frac{396350970581}{157242193266}a+\frac{13789344321}{26207032211}$, $\frac{2533313}{13418000492032}a^{11}-\frac{1200820687}{40254001476096}a^{10}+\frac{1020522087}{13418000492032}a^{9}-\frac{35792156997}{13418000492032}a^{8}+\frac{428526299117}{40254001476096}a^{7}-\frac{4683429685895}{20127000738048}a^{6}-\frac{17347301423}{38410306752}a^{5}-\frac{15366074947911}{838625030752}a^{4}-\frac{1083143354067}{104828128844}a^{3}-\frac{19\!\cdots\!19}{1257937546128}a^{2}+\frac{434466322207873}{628968773064}a-\frac{12862487208612}{26207032211}$ 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:  \( 135561.594855 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 135561.594855 \cdot 1443}{10\cdot\sqrt{39548330211803192814453125}}\cr\approx \mathstrut & 0.191389511957 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416)
 
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 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416, 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 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416);
 
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 + 87*x^10 - 221*x^9 + 7751*x^8 + 27358*x^7 + 649836*x^6 + 1330904*x^5 + 53241808*x^4 + 30023808*x^3 + 16929792*x^2 + 9510912*x + 5308416);
 
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

\(\Q(\sqrt{5}) \), 3.3.67081.1, \(\Q(\zeta_{5})\), 6.6.562482570125.1

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 ${\href{/padicField/2.4.0.1}{4} }^{3}$ ${\href{/padicField/3.4.0.1}{4} }^{3}$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}$ ${\href{/padicField/13.12.0.1}{12} }$ ${\href{/padicField/17.12.0.1}{12} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.12.0.1}{12} }$ ${\href{/padicField/29.2.0.1}{2} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ R ${\href{/padicField/41.3.0.1}{3} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{3}$ ${\href{/padicField/47.12.0.1}{12} }$ ${\href{/padicField/53.12.0.1}{12} }$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 7.12.8.2$x^{12} - 70 x^{9} + 1519 x^{6} - 4802 x^{3} + 21609$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
\(37\) Copy content Toggle raw display 37.12.8.3$x^{12} - 444 x^{9} + 54760 x^{6} + 27960456 x^{3} + 7496644$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$