Properties

Label 12.0.164...352.1
Degree 1212
Signature [0,6][0, 6]
Discriminant 1.647×10241.647\times 10^{24}
Root discriminant 104.24104.24
Ramified primes 2,612,61
Class number 349349 (GRH)
Class group [349] (GRH)
Galois group C12C_{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 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057)
 
gp: K = bnfinit(y^12 - 4*y^11 - 62*y^10 + 232*y^9 + 1679*y^8 - 5668*y^7 - 20398*y^6 + 62556*y^5 + 37242*y^4 - 194132*y^3 + 1081520*y^2 - 893952*y + 689057, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057)
 

x124x1162x10+232x9+1679x85668x720398x6+62556x5++689057 x^{12} - 4 x^{11} - 62 x^{10} + 232 x^{9} + 1679 x^{8} - 5668 x^{7} - 20398 x^{6} + 62556 x^{5} + \cdots + 689057 Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  1212
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [0,6][0, 6]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   16467532794547152638443521646753279454715263844352 =233618\medspace = 2^{33}\cdot 61^{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:  104.24104.24
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:  211/4612/3104.244314566085582^{11/4}61^{2/3}\approx 104.24431456608558
Ramified primes:   22, 6161 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(2)\Q(\sqrt{2})
Aut(K/Q)\Aut(K/\Q) == Gal(K/Q)\Gal(K/\Q):   C12C_{12}
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over Q\Q.
Conductor:  976=2461976=2^{4}\cdot 61
Dirichlet character group:    {\lbraceχ976(1,)\chi_{976}(1,·), χ976(611,)\chi_{976}(611,·), χ976(257,)\chi_{976}(257,·), χ976(379,)\chi_{976}(379,·), χ976(169,)\chi_{976}(169,·), χ976(779,)\chi_{976}(779,·), χ976(291,)\chi_{976}(291,·), χ976(657,)\chi_{976}(657,·), χ976(867,)\chi_{976}(867,·), χ976(489,)\chi_{976}(489,·), χ976(745,)\chi_{976}(745,·), χ976(123,)\chi_{976}(123,·)}\rbrace
This is a CM field.
Reflex fields:  4.0.2048.22^{2}, 12.0.1646753279454715263844352.130^{30}

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, a7a^{7}, 13a8+13a713a6+13a5+13a413a313\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}, 13a9+13a713a6+13a4+13a313a+13\frac{1}{3}a^{9}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a+\frac{1}{3}, 17402519347a10+2704076267402519347a9+9463497717402519347a8+12399417887402519347a735914685037402519347a611634015467402519347a5+11881250902467506449a410459512742467506449a312159300497402519347a2+6868876602467506449a11564311182467506449\frac{1}{7402519347}a^{10}+\frac{270407626}{7402519347}a^{9}+\frac{946349771}{7402519347}a^{8}+\frac{1239941788}{7402519347}a^{7}-\frac{3591468503}{7402519347}a^{6}-\frac{1163401546}{7402519347}a^{5}+\frac{1188125090}{2467506449}a^{4}-\frac{1045951274}{2467506449}a^{3}-\frac{1215930049}{7402519347}a^{2}+\frac{686887660}{2467506449}a-\frac{1156431118}{2467506449}, 134 ⁣ ⁣11a11103359034734 ⁣ ⁣11a10+51 ⁣ ⁣1711 ⁣ ⁣37a931 ⁣ ⁣0211 ⁣ ⁣37a816 ⁣ ⁣8634 ⁣ ⁣11a7+14 ⁣ ⁣2634 ⁣ ⁣11a6+17 ⁣ ⁣0434 ⁣ ⁣11a5+10 ⁣ ⁣3111 ⁣ ⁣37a416 ⁣ ⁣2734 ⁣ ⁣11a3+21 ⁣ ⁣5511 ⁣ ⁣37a252 ⁣ ⁣1534 ⁣ ⁣11a24 ⁣ ⁣9315 ⁣ ⁣57\frac{1}{34\!\cdots\!11}a^{11}-\frac{1033590347}{34\!\cdots\!11}a^{10}+\frac{51\!\cdots\!17}{11\!\cdots\!37}a^{9}-\frac{31\!\cdots\!02}{11\!\cdots\!37}a^{8}-\frac{16\!\cdots\!86}{34\!\cdots\!11}a^{7}+\frac{14\!\cdots\!26}{34\!\cdots\!11}a^{6}+\frac{17\!\cdots\!04}{34\!\cdots\!11}a^{5}+\frac{10\!\cdots\!31}{11\!\cdots\!37}a^{4}-\frac{16\!\cdots\!27}{34\!\cdots\!11}a^{3}+\frac{21\!\cdots\!55}{11\!\cdots\!37}a^{2}-\frac{52\!\cdots\!15}{34\!\cdots\!11}a-\frac{24\!\cdots\!93}{15\!\cdots\!57} Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

C349C_{349}, which has order 349349 (assuming GRH)

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

Relative class number: 349349 (assuming GRH)

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  55
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) 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:   2442467506449a11+26102467506449a10280982467506449a92494412467506449a8+13364402467506449a7+98234182467506449a6317585682467506449a51765355872467506449a4+3477260162467506449a3+11857007152467506449a211984906242467506449a+22044255122467506449\frac{244}{2467506449}a^{11}+\frac{2610}{2467506449}a^{10}-\frac{28098}{2467506449}a^{9}-\frac{249441}{2467506449}a^{8}+\frac{1336440}{2467506449}a^{7}+\frac{9823418}{2467506449}a^{6}-\frac{31758568}{2467506449}a^{5}-\frac{176535587}{2467506449}a^{4}+\frac{347726016}{2467506449}a^{3}+\frac{1185700715}{2467506449}a^{2}-\frac{1198490624}{2467506449}a+\frac{2204425512}{2467506449}, 1650898483348434 ⁣ ⁣11a1159877132249590534 ⁣ ⁣11a10+12 ⁣ ⁣1634 ⁣ ⁣11a9+36 ⁣ ⁣5734 ⁣ ⁣11a810 ⁣ ⁣3234 ⁣ ⁣11a710 ⁣ ⁣8134 ⁣ ⁣11a6+26 ⁣ ⁣9834 ⁣ ⁣11a5+46 ⁣ ⁣6111 ⁣ ⁣37a410 ⁣ ⁣4411 ⁣ ⁣37a325 ⁣ ⁣5011 ⁣ ⁣37a2+78 ⁣ ⁣7634 ⁣ ⁣11a+95 ⁣ ⁣5815 ⁣ ⁣57\frac{16508984833484}{34\!\cdots\!11}a^{11}-\frac{598771322495905}{34\!\cdots\!11}a^{10}+\frac{12\!\cdots\!16}{34\!\cdots\!11}a^{9}+\frac{36\!\cdots\!57}{34\!\cdots\!11}a^{8}-\frac{10\!\cdots\!32}{34\!\cdots\!11}a^{7}-\frac{10\!\cdots\!81}{34\!\cdots\!11}a^{6}+\frac{26\!\cdots\!98}{34\!\cdots\!11}a^{5}+\frac{46\!\cdots\!61}{11\!\cdots\!37}a^{4}-\frac{10\!\cdots\!44}{11\!\cdots\!37}a^{3}-\frac{25\!\cdots\!50}{11\!\cdots\!37}a^{2}+\frac{78\!\cdots\!76}{34\!\cdots\!11}a+\frac{95\!\cdots\!58}{15\!\cdots\!57}, 26951240244450 ⁣ ⁣19a11+2996745717377915 ⁣ ⁣57a1017235949579849615 ⁣ ⁣57a962549243226091350 ⁣ ⁣19a8+80 ⁣ ⁣5615 ⁣ ⁣57a7+17 ⁣ ⁣8150 ⁣ ⁣19a654 ⁣ ⁣7050 ⁣ ⁣19a568 ⁣ ⁣8715 ⁣ ⁣57a4+14 ⁣ ⁣3615 ⁣ ⁣57a3+23 ⁣ ⁣9415 ⁣ ⁣57a289 ⁣ ⁣5250 ⁣ ⁣19a+28 ⁣ ⁣7615 ⁣ ⁣57\frac{269512402444}{50\!\cdots\!19}a^{11}+\frac{29967457173779}{15\!\cdots\!57}a^{10}-\frac{172359495798496}{15\!\cdots\!57}a^{9}-\frac{625492432260913}{50\!\cdots\!19}a^{8}+\frac{80\!\cdots\!56}{15\!\cdots\!57}a^{7}+\frac{17\!\cdots\!81}{50\!\cdots\!19}a^{6}-\frac{54\!\cdots\!70}{50\!\cdots\!19}a^{5}-\frac{68\!\cdots\!87}{15\!\cdots\!57}a^{4}+\frac{14\!\cdots\!36}{15\!\cdots\!57}a^{3}+\frac{23\!\cdots\!94}{15\!\cdots\!57}a^{2}-\frac{89\!\cdots\!52}{50\!\cdots\!19}a+\frac{28\!\cdots\!76}{15\!\cdots\!57}, 7872130963809234 ⁣ ⁣11a11+32059418656232234 ⁣ ⁣11a1061 ⁣ ⁣9634 ⁣ ⁣11a923 ⁣ ⁣2434 ⁣ ⁣11a8+65 ⁣ ⁣4811 ⁣ ⁣37a7+70 ⁣ ⁣3334 ⁣ ⁣11a630 ⁣ ⁣0434 ⁣ ⁣11a510 ⁣ ⁣2634 ⁣ ⁣11a4+19 ⁣ ⁣5234 ⁣ ⁣11a3+44 ⁣ ⁣9734 ⁣ ⁣11a248 ⁣ ⁣8234 ⁣ ⁣11a+67 ⁣ ⁣8150 ⁣ ⁣19\frac{78721309638092}{34\!\cdots\!11}a^{11}+\frac{320594186562322}{34\!\cdots\!11}a^{10}-\frac{61\!\cdots\!96}{34\!\cdots\!11}a^{9}-\frac{23\!\cdots\!24}{34\!\cdots\!11}a^{8}+\frac{65\!\cdots\!48}{11\!\cdots\!37}a^{7}+\frac{70\!\cdots\!33}{34\!\cdots\!11}a^{6}-\frac{30\!\cdots\!04}{34\!\cdots\!11}a^{5}-\frac{10\!\cdots\!26}{34\!\cdots\!11}a^{4}+\frac{19\!\cdots\!52}{34\!\cdots\!11}a^{3}+\frac{44\!\cdots\!97}{34\!\cdots\!11}a^{2}-\frac{48\!\cdots\!82}{34\!\cdots\!11}a+\frac{67\!\cdots\!81}{50\!\cdots\!19}, 3348577353844434 ⁣ ⁣11a11+46805486642804234 ⁣ ⁣11a1035 ⁣ ⁣8834 ⁣ ⁣11a930 ⁣ ⁣5434 ⁣ ⁣11a8+45 ⁣ ⁣9211 ⁣ ⁣37a7+85 ⁣ ⁣3734 ⁣ ⁣11a626 ⁣ ⁣1634 ⁣ ⁣11a511 ⁣ ⁣5834 ⁣ ⁣11a4+23 ⁣ ⁣7234 ⁣ ⁣11a3+41 ⁣ ⁣0534 ⁣ ⁣11a246 ⁣ ⁣3834 ⁣ ⁣11a+61 ⁣ ⁣8350 ⁣ ⁣19\frac{33485773538444}{34\!\cdots\!11}a^{11}+\frac{468054866428042}{34\!\cdots\!11}a^{10}-\frac{35\!\cdots\!88}{34\!\cdots\!11}a^{9}-\frac{30\!\cdots\!54}{34\!\cdots\!11}a^{8}+\frac{45\!\cdots\!92}{11\!\cdots\!37}a^{7}+\frac{85\!\cdots\!37}{34\!\cdots\!11}a^{6}-\frac{26\!\cdots\!16}{34\!\cdots\!11}a^{5}-\frac{11\!\cdots\!58}{34\!\cdots\!11}a^{4}+\frac{23\!\cdots\!72}{34\!\cdots\!11}a^{3}+\frac{41\!\cdots\!05}{34\!\cdots\!11}a^{2}-\frac{46\!\cdots\!38}{34\!\cdots\!11}a+\frac{61\!\cdots\!83}{50\!\cdots\!19} 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:  27056.6931186 27056.6931186 (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(20(2π)627056.693118634921646753279454715263844352(0.226378453531 \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 27056.6931186 \cdot 349}{2\cdot\sqrt{1646753279454715263844352}}\cr\approx \mathstrut & 0.226378453531 \end{aligned} (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057)
 
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 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057, 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(Rationals()); K<a> := NumberField(x^12 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057);
 
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 - 4*x^11 - 62*x^10 + 232*x^9 + 1679*x^8 - 5668*x^7 - 20398*x^6 + 62556*x^5 + 37242*x^4 - 194132*x^3 + 1081520*x^2 - 893952*x + 689057);
 
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

C12C_{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 C12C_{12}
Character table for C12C_{12}

Intermediate fields

Q(2)\Q(\sqrt{2}) , 3.3.3721.1, 4.0.2048.2, 6.6.7089070592.1

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

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type R 43{\href{/padicField/3.4.0.1}{4} }^{3} 12{\href{/padicField/5.12.0.1}{12} } 34{\href{/padicField/7.3.0.1}{3} }^{4} 43{\href{/padicField/11.4.0.1}{4} }^{3} 12{\href{/padicField/13.12.0.1}{12} } 34{\href{/padicField/17.3.0.1}{3} }^{4} 12{\href{/padicField/19.12.0.1}{12} } 112{\href{/padicField/23.1.0.1}{1} }^{12} 12{\href{/padicField/29.12.0.1}{12} } 62{\href{/padicField/31.6.0.1}{6} }^{2} 43{\href{/padicField/37.4.0.1}{4} }^{3} 26{\href{/padicField/41.2.0.1}{2} }^{6} 12{\href{/padicField/43.12.0.1}{12} } 62{\href{/padicField/47.6.0.1}{6} }^{2} 43{\href{/padicField/53.4.0.1}{4} }^{3} 12{\href{/padicField/59.12.0.1}{12} }

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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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

ppLabelPolynomial ee ff cc Galois group Slope content
22 Copy content Toggle raw display 2.3.4.33a1.177x12+4x10+4x9+6x8+12x7+14x6+12x5+21x4+16x3+10x2+12x+7x^{12} + 4 x^{10} + 4 x^{9} + 6 x^{8} + 12 x^{7} + 14 x^{6} + 12 x^{5} + 21 x^{4} + 16 x^{3} + 10 x^{2} + 12 x + 744333333C12C_{12}[3,4]3[3, 4]^{3}
6161 Copy content Toggle raw display 61.4.3.8a1.3x12+9x10+120x9+33x8+720x7+4863x6+1560x5+14466x4+65440x3+9636x2+480x+69x^{12} + 9 x^{10} + 120 x^{9} + 33 x^{8} + 720 x^{7} + 4863 x^{6} + 1560 x^{5} + 14466 x^{4} + 65440 x^{3} + 9636 x^{2} + 480 x + 69334488C12C_{12}[ ]34[\ ]_{3}^{4}

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)