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Properties

Label 12.4.74049191673856.3
Degree $12$
Signature $[4, 4]$
Discriminant $7.405\times 10^{13}$
Root discriminant \(14.32\)
Ramified primes $2,7$
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 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27)
 
gp: K = bnfinit(y^12 - 2*y^11 - 2*y^10 + 16*y^9 - 13*y^8 - 28*y^7 + 50*y^6 - 62*y^4 + 26*y^3 + 12*y^2 - 54*y - 27, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27)
 

\( x^{12} - 2x^{11} - 2x^{10} + 16x^{9} - 13x^{8} - 28x^{7} + 50x^{6} - 62x^{4} + 26x^{3} + 12x^{2} - 54x - 27 \) 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:   \(74049191673856\) \(\medspace = 2^{18}\cdot 7^{10}\) 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:  \(14.32\)
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:  $2^{7/4}7^{5/6}\approx 17.023578553967216$
Ramified primes:   \(2\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}$, $\frac{1}{195309}a^{11}+\frac{20347}{195309}a^{10}-\frac{13979}{195309}a^{9}-\frac{23648}{195309}a^{8}+\frac{28211}{195309}a^{7}-\frac{77746}{195309}a^{6}+\frac{14699}{195309}a^{5}+\frac{8923}{65103}a^{4}-\frac{59585}{195309}a^{3}-\frac{16867}{195309}a^{2}+\frac{20516}{65103}a-\frac{8491}{21701}$ 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:   $\frac{101939}{195309}a^{11}-\frac{289159}{195309}a^{10}+\frac{34286}{195309}a^{9}+\frac{1607987}{195309}a^{8}-\frac{2662913}{195309}a^{7}-\frac{686819}{195309}a^{6}+\frac{5719777}{195309}a^{5}-\frac{1558190}{65103}a^{4}-\frac{2659741}{195309}a^{3}+\frac{4912945}{195309}a^{2}-\frac{856288}{65103}a-\frac{388581}{21701}$, $\frac{55070}{195309}a^{11}-\frac{173752}{195309}a^{10}+\frac{84548}{195309}a^{9}+\frac{806288}{195309}a^{8}-\frac{1665797}{195309}a^{7}+\frac{286987}{195309}a^{6}+\frac{2652451}{195309}a^{5}-\frac{1049482}{65103}a^{4}-\frac{154750}{195309}a^{3}+\frac{1977004}{195309}a^{2}-\frac{567269}{65103}a-\frac{138129}{21701}$, $\frac{6968}{65103}a^{11}-\frac{16438}{65103}a^{10}-\frac{11584}{65103}a^{9}+\frac{126635}{65103}a^{8}-\frac{48439}{21701}a^{7}-\frac{61891}{21701}a^{6}+\frac{514736}{65103}a^{5}-\frac{84705}{21701}a^{4}-\frac{146256}{21701}a^{3}+\frac{174728}{21701}a^{2}-\frac{205708}{65103}a-\frac{111890}{21701}$, $\frac{82406}{195309}a^{11}-\frac{273295}{195309}a^{10}+\frac{174317}{195309}a^{9}+\frac{1162865}{195309}a^{8}-\frac{2676584}{195309}a^{7}+\frac{960796}{195309}a^{6}+\frac{3625144}{195309}a^{5}-\frac{1787428}{65103}a^{4}+\frac{1208810}{195309}a^{3}+\frac{2350756}{195309}a^{2}-\frac{1016557}{65103}a-\frac{112508}{21701}$, $\frac{236113}{195309}a^{11}-\frac{670601}{195309}a^{10}+\frac{98473}{195309}a^{9}+\frac{3669442}{195309}a^{8}-\frac{6149284}{195309}a^{7}-\frac{1310860}{195309}a^{6}+\frac{12679142}{195309}a^{5}-\frac{1195317}{21701}a^{4}-\frac{5047736}{195309}a^{3}+\frac{10183916}{195309}a^{2}-\frac{2021105}{65103}a-\frac{791535}{21701}$, $\frac{299072}{195309}a^{11}-\frac{870868}{195309}a^{10}+\frac{187172}{195309}a^{9}+\frac{4632062}{195309}a^{8}-\frac{8106071}{195309}a^{7}-\frac{1091807}{195309}a^{6}+\frac{16059694}{195309}a^{5}-\frac{1618980}{21701}a^{4}-\frac{5420200}{195309}a^{3}+\frac{12818719}{195309}a^{2}-\frac{2692213}{65103}a-\frac{967578}{21701}$, $\frac{20896}{21701}a^{11}-\frac{60183}{21701}a^{10}+\frac{11977}{21701}a^{9}+\frac{969433}{65103}a^{8}-\frac{1681103}{65103}a^{7}-\frac{239173}{65103}a^{6}+\frac{1122802}{21701}a^{5}-\frac{3103099}{65103}a^{4}-\frac{1108307}{65103}a^{3}+\frac{2822158}{65103}a^{2}-\frac{632086}{21701}a-\frac{612668}{21701}$ 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:  \( 185.388043669 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{4}\cdot 185.388043669 \cdot 1}{2\cdot\sqrt{74049191673856}}\cr\approx \mathstrut & 0.268615470077 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27)
 
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 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27, 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 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27);
 
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 - 2*x^11 - 2*x^10 + 16*x^9 - 13*x^8 - 28*x^7 + 50*x^6 - 62*x^4 + 26*x^3 + 12*x^2 - 54*x - 27);
 
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

\(\Q(\sqrt{7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{28})^+\), 6.2.1075648.1, 6.2.153664.1

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: 6.2.1075648.1
Degree 8 sibling: 8.0.1927561216.9
Degree 12 sibling: 12.0.74049191673856.4
Minimal sibling: 6.2.1075648.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.3.0.1}{3} }^{4}$ ${\href{/padicField/5.6.0.1}{6} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{4}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.3.0.1}{3} }^{4}$ ${\href{/padicField/37.3.0.1}{3} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.3.0.1}{3} }^{4}$ ${\href{/padicField/53.3.0.1}{3} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{4}$

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
\(2\) Copy content Toggle raw display 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}$
\(7\) Copy content Toggle raw display 7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} + 42$$6$$1$$5$$C_6$$[\ ]_{6}$