Properties

Label 12.4.3429038075724121.1
Degree $12$
Signature $[4, 4]$
Discriminant $3.429\times 10^{15}$
Root discriminant \(19.71\)
Ramified primes $7,29$
Class number $1$
Class group trivial
Galois group $A_4 \times C_2$ (as 12T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169)
 
gp: K = bnfinit(y^12 - 3*y^11 - 2*y^10 + y^9 + 36*y^8 - 23*y^7 - 116*y^6 - 10*y^5 + 319*y^4 + 209*y^3 - 473*y^2 - 247*y + 169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169)
 

\( x^{12} - 3 x^{11} - 2 x^{10} + x^{9} + 36 x^{8} - 23 x^{7} - 116 x^{6} - 10 x^{5} + 319 x^{4} + 209 x^{3} - 473 x^{2} - 247 x + 169 \) 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:  $[4, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(3429038075724121\) \(\medspace = 7^{8}\cdot 29^{6}\) 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:  \(19.71\)
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:  $7^{2/3}29^{1/2}\approx 19.705964328162043$
Ramified primes:   \(7\), \(29\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3251854344479}a^{11}+\frac{1339303801404}{3251854344479}a^{10}-\frac{650708486781}{3251854344479}a^{9}+\frac{444497570622}{3251854344479}a^{8}+\frac{737894480411}{3251854344479}a^{7}-\frac{1601948339807}{3251854344479}a^{6}-\frac{377725152648}{3251854344479}a^{5}+\frac{199110752051}{3251854344479}a^{4}-\frac{10086624021}{3251854344479}a^{3}+\frac{774023811861}{3251854344479}a^{2}+\frac{1460316099341}{3251854344479}a+\frac{5171728660}{250142641883}$ 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:  $7$
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:   $\frac{56106389219}{3251854344479}a^{11}-\frac{174958974662}{3251854344479}a^{10}-\frac{340210787}{3251854344479}a^{9}-\frac{166765475269}{3251854344479}a^{8}+\frac{1891909408047}{3251854344479}a^{7}-\frac{1910844730960}{3251854344479}a^{6}-\frac{3434369672867}{3251854344479}a^{5}-\frac{1268405889253}{3251854344479}a^{4}+\frac{11789137748016}{3251854344479}a^{3}+\frac{4339535272797}{3251854344479}a^{2}-\frac{13034481263451}{3251854344479}a+\frac{161219520514}{250142641883}$, $\frac{58803426484}{3251854344479}a^{11}-\frac{177405663322}{3251854344479}a^{10}-\frac{23785815580}{3251854344479}a^{9}-\frac{179420819568}{3251854344479}a^{8}+\frac{2018636976053}{3251854344479}a^{7}-\frac{1734339135538}{3251854344479}a^{6}-\frac{3972636236416}{3251854344479}a^{5}-\frac{2370330546346}{3251854344479}a^{4}+\frac{13045666786482}{3251854344479}a^{3}+\frac{7373138608716}{3251854344479}a^{2}-\frac{12581484433854}{3251854344479}a-\frac{102922591481}{250142641883}$, $\frac{431095498}{250142641883}a^{11}-\frac{177614125}{250142641883}a^{10}-\frac{4145848903}{250142641883}a^{9}-\frac{1710346819}{250142641883}a^{8}+\frac{15686413085}{250142641883}a^{7}+\frac{20579678181}{250142641883}a^{6}-\frac{54703212980}{250142641883}a^{5}-\frac{101172304383}{250142641883}a^{4}+\frac{127606949367}{250142641883}a^{3}+\frac{268309536382}{250142641883}a^{2}+\frac{31207780279}{250142641883}a+\frac{68451774258}{250142641883}$, $\frac{29199783557}{3251854344479}a^{11}-\frac{70702201935}{3251854344479}a^{10}-\frac{65016107863}{3251854344479}a^{9}-\frac{137140237082}{3251854344479}a^{8}+\frac{1071775580940}{3251854344479}a^{7}-\frac{187240917460}{3251854344479}a^{6}-\frac{2502127258492}{3251854344479}a^{5}-\frac{3259607601964}{3251854344479}a^{4}+\frac{6121005968413}{3251854344479}a^{3}+\frac{11359356968904}{3251854344479}a^{2}-\frac{6552911658000}{3251854344479}a-\frac{697586998188}{250142641883}$, $\frac{205014965034}{3251854344479}a^{11}-\frac{535665092951}{3251854344479}a^{10}-\frac{237240887148}{3251854344479}a^{9}-\frac{861903657671}{3251854344479}a^{8}+\frac{6534104707766}{3251854344479}a^{7}-\frac{3671876923012}{3251854344479}a^{6}-\frac{13237724533635}{3251854344479}a^{5}-\frac{13742891253922}{3251854344479}a^{4}+\frac{35386394199856}{3251854344479}a^{3}+\frac{31137407686446}{3251854344479}a^{2}-\frac{20400363329846}{3251854344479}a+\frac{17331678044}{250142641883}$, $\frac{58563646917}{3251854344479}a^{11}-\frac{258755355119}{3251854344479}a^{10}+\frac{233374829215}{3251854344479}a^{9}-\frac{216916466069}{3251854344479}a^{8}+\frac{2362712089399}{3251854344479}a^{7}-\frac{4484934294294}{3251854344479}a^{6}-\frac{1198584091202}{3251854344479}a^{5}+\frac{2365649789894}{3251854344479}a^{4}+\frac{15969062053618}{3251854344479}a^{3}-\frac{7441563958365}{3251854344479}a^{2}-\frac{22014745962241}{3251854344479}a+\frac{681767401841}{250142641883}$, $\frac{18164859367}{3251854344479}a^{11}-\frac{57191615366}{3251854344479}a^{10}-\frac{33883030074}{3251854344479}a^{9}+\frac{41610464160}{3251854344479}a^{8}+\frac{666590281511}{3251854344479}a^{7}-\frac{544519333447}{3251854344479}a^{6}-\frac{2283629281994}{3251854344479}a^{5}+\frac{356617969879}{3251854344479}a^{4}+\frac{6896514795166}{3251854344479}a^{3}+\frac{2539926569237}{3251854344479}a^{2}-\frac{14877436160989}{3251854344479}a-\frac{880263521708}{250142641883}$ 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:  \( 1824.19500337 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 1824.19500337 \cdot 1}{2\cdot\sqrt{3429038075724121}}\cr\approx \mathstrut & 0.388413725663 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169)
 
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 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169, 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 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169);
 
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 - 3*x^11 - 2*x^10 + x^9 + 36*x^8 - 23*x^7 - 116*x^6 - 10*x^5 + 319*x^4 + 209*x^3 - 473*x^2 - 247*x + 169);
 
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

$C_2\times A_4$ (as 12T7):

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 24
The 8 conjugacy class representatives for $A_4 \times C_2$
Character table for $A_4 \times C_2$

Intermediate fields

\(\Q(\sqrt{29}) \), \(\Q(\zeta_{7})^+\), 6.2.69629.1, 6.6.58557989.1, 6.2.2019241.1

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

Galois closure: deg 24
Degree 6 sibling: 6.2.69629.1
Degree 8 sibling: 8.0.1698181681.2
Degree 12 sibling: 12.0.4077334216081.1
Minimal sibling: 6.2.69629.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.6.0.1}{6} }^{2}$ ${\href{/padicField/3.6.0.1}{6} }^{2}$ ${\href{/padicField/5.3.0.1}{3} }^{4}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.3.0.1}{3} }^{4}$ R ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{2}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(7\) Copy content Toggle raw display 7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} + 7$$3$$1$$2$$C_3$$[\ ]_{3}$
\(29\) Copy content Toggle raw display 29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.1$x^{2} + 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$