Properties

Label 12.0.289254654976000000.2
Degree $12$
Signature $[0, 6]$
Discriminant $2.893\times 10^{17}$
Root discriminant \(28.52\)
Ramified primes $2,5,7$
Class number $6$
Class group [6]
Galois group $C_6\times S_3$ (as 12T18)

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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 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484)
 
gp: K = bnfinit(y^12 - 2*y^11 + 2*y^10 - 14*y^9 - 60*y^8 + 36*y^7 + 146*y^6 + 268*y^5 + 936*y^4 + 1316*y^3 + 1152*y^2 + 1056*y + 484, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 2*x^11 + 2*x^10 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 2*x^11 + 2*x^10 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484)
 

\( x^{12} - 2 x^{11} + 2 x^{10} - 14 x^{9} - 60 x^{8} + 36 x^{7} + 146 x^{6} + 268 x^{5} + 936 x^{4} + 1316 x^{3} + 1152 x^{2} + 1056 x + 484 \) 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:   \(289254654976000000\) \(\medspace = 2^{16}\cdot 5^{6}\cdot 7^{10}\) 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:  \(28.52\)
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:  $2^{4/3}5^{1/2}7^{5/6}\approx 28.517187846962024$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $6$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{246}a^{10}+\frac{5}{82}a^{9}+\frac{17}{82}a^{8}-\frac{23}{246}a^{7}-\frac{5}{123}a^{6}-\frac{35}{123}a^{5}+\frac{16}{123}a^{4}-\frac{10}{123}a^{3}-\frac{46}{123}a^{2}-\frac{32}{123}a+\frac{37}{123}$, $\frac{1}{29019206238}a^{11}-\frac{772973}{707785518}a^{10}+\frac{383066292}{4836534373}a^{9}+\frac{3500025368}{14509603119}a^{8}-\frac{1660023802}{14509603119}a^{7}-\frac{1387596033}{9673068746}a^{6}+\frac{985753764}{4836534373}a^{5}+\frac{6747052369}{14509603119}a^{4}-\frac{512346393}{4836534373}a^{3}-\frac{3698711227}{14509603119}a^{2}+\frac{1539225220}{4836534373}a-\frac{448759106}{1319054829}$ 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

$C_{6}$, which has order $6$

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:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{10782023}{235928506} a^{11} - \frac{16125900}{117964253} a^{10} + \frac{53426107}{235928506} a^{9} - \frac{101677030}{117964253} a^{8} - \frac{222652114}{117964253} a^{7} + \frac{415754024}{117964253} a^{6} + \frac{380392390}{117964253} a^{5} + \frac{1051946014}{117964253} a^{4} + \frac{3983952406}{117964253} a^{3} + \frac{3102320718}{117964253} a^{2} + \frac{2966020988}{117964253} a + \frac{237351277}{10724023} \)  (order $4$) 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:   $\frac{262874185}{9673068746}a^{11}-\frac{2094867347}{29019206238}a^{10}+\frac{492483235}{4836534373}a^{9}-\frac{2160566125}{4836534373}a^{8}-\frac{39056968607}{29019206238}a^{7}+\frac{27457463152}{14509603119}a^{6}+\frac{38533527514}{14509603119}a^{5}+\frac{81835091053}{14509603119}a^{4}+\frac{318267823217}{14509603119}a^{3}+\frac{301460933069}{14509603119}a^{2}+\frac{256192566247}{14509603119}a+\frac{22669828907}{1319054829}$, $\frac{1731886447}{29019206238}a^{11}-\frac{5279801845}{29019206238}a^{10}+\frac{1512013369}{4836534373}a^{9}-\frac{16813131667}{14509603119}a^{8}-\frac{69217927397}{29019206238}a^{7}+\frac{22523784148}{4836534373}a^{6}+\frac{17169871518}{4836534373}a^{5}+\frac{173091388810}{14509603119}a^{4}+\frac{214312132781}{4836534373}a^{3}+\frac{481217776601}{14509603119}a^{2}+\frac{172632266711}{4836534373}a+\frac{42128465905}{1319054829}$, $\frac{7177145}{879369886}a^{11}-\frac{11074926}{439684943}a^{10}+\frac{17849810}{439684943}a^{9}-\frac{64488424}{439684943}a^{8}-\frac{146020517}{439684943}a^{7}+\frac{289587598}{439684943}a^{6}+\frac{290480850}{439684943}a^{5}+\frac{311984746}{439684943}a^{4}+\frac{2759931068}{439684943}a^{3}+\frac{2242681759}{439684943}a^{2}+\frac{2202407262}{439684943}a+\frac{2429805599}{439684943}$, $\frac{54728438}{1319054829}a^{11}-\frac{150396872}{1319054829}a^{10}+\frac{145422357}{879369886}a^{9}-\frac{1825854227}{2638109658}a^{8}-\frac{2648272630}{1319054829}a^{7}+\frac{33244078}{10724023}a^{6}+\frac{1638679288}{439684943}a^{5}+\frac{10742045998}{1319054829}a^{4}+\frac{14584880891}{439684943}a^{3}+\frac{37722655517}{1319054829}a^{2}+\frac{10988568609}{439684943}a+\frac{32338435295}{1319054829}$, $\frac{229625801}{14509603119}a^{11}-\frac{226068915}{4836534373}a^{10}+\frac{366794995}{4836534373}a^{9}-\frac{8617746983}{29019206238}a^{8}-\frac{18909010837}{29019206238}a^{7}+\frac{17308489918}{14509603119}a^{6}+\frac{18109201258}{14509603119}a^{5}+\frac{1125280342}{353892759}a^{4}+\frac{159518035202}{14509603119}a^{3}+\frac{126070780954}{14509603119}a^{2}+\frac{129773268796}{14509603119}a+\frac{3206618023}{439684943}$ 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:  \( 1815.0374540202486 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{6}\cdot 1815.0374540202486 \cdot 6}{4\cdot\sqrt{289254654976000000}}\cr\approx \mathstrut & 0.311469755618998 \end{aligned}\]

# 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 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484)
 
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 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484, 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 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484);
 
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 - 14*x^9 - 60*x^8 + 36*x^7 + 146*x^6 + 268*x^5 + 936*x^4 + 1316*x^3 + 1152*x^2 + 1056*x + 484);
 
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

$C_6\times S_3$ (as 12T18):

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 36
The 18 conjugacy class representatives for $C_6\times S_3$
Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{35})\), 6.0.33614000.1

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 36
Degree 18 siblings: 18.6.155568095557812224000000000.1, 18.0.177792109208928256000000.1
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 ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}$ R R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{4}$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{2}$ ${\href{/padicField/23.6.0.1}{6} }^{2}$ ${\href{/padicField/29.3.0.1}{3} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{6}$ ${\href{/padicField/43.2.0.1}{2} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.6.0.1}{6} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}$

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
\(2\) Copy content Toggle raw display 2.12.16.3$x^{12} + 4 x^{11} + 8 x^{10} + 10 x^{9} + 8 x^{8} + 4 x^{7} + 2 x^{6} - 4 x^{5} - 4 x^{4} + 4 x^{3} + 4$$6$$2$$16$$C_6\times S_3$$[2]_{3}^{6}$
\(5\) Copy content Toggle raw display 5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} + 75 x^{2} - 375$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(7\) Copy content Toggle raw display 7.12.10.1$x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.35.2t1.a.a$1$ $ 5 \cdot 7 $ \(\Q(\sqrt{-35}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.140.2t1.a.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ \(\Q(\sqrt{35}) \) $C_2$ (as 2T1) $1$ $1$
1.28.6t1.a.a$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.140.6t1.a.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.6.134456000.1 $C_6$ (as 6T1) $0$ $1$
1.35.6t1.a.a$1$ $ 5 \cdot 7 $ 6.0.2100875.1 $C_6$ (as 6T1) $0$ $-1$
1.28.6t1.a.b$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.35.6t1.a.b$1$ $ 5 \cdot 7 $ 6.0.2100875.1 $C_6$ (as 6T1) $0$ $-1$
1.140.6t1.a.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.6.134456000.1 $C_6$ (as 6T1) $0$ $1$
2.140.3t2.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 3.1.140.1 $S_3$ (as 3T2) $1$ $0$
2.560.6t3.b.a$2$ $ 2^{4} \cdot 5 \cdot 7 $ 6.0.313600.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.3920.12t18.b.a$2$ $ 2^{4} \cdot 5 \cdot 7^{2}$ 12.0.289254654976000000.2 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.980.6t5.a.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 6.0.33614000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.3920.12t18.b.b$2$ $ 2^{4} \cdot 5 \cdot 7^{2}$ 12.0.289254654976000000.2 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.980.6t5.a.b$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 6.0.33614000.1 $S_3\times C_3$ (as 6T5) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.