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Properties

Label 28.0.159...112.1
Degree $28$
Signature $[0, 14]$
Discriminant $1.591\times 10^{61}$
Root discriminant \(153.38\)
Ramified primes $2,29$
Class number not computed
Class group not computed
Galois group $C_{28}$ (as 28T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648)
 
gp: K = bnfinit(y^28 + 116*y^26 + 5162*y^24 + 117160*y^22 + 1511944*y^20 + 11760544*y^18 + 56876888*y^16 + 173227904*y^14 + 331475104*y^12 + 392753728*y^10 + 281553344*y^8 + 117551616*y^6 + 26535232*y^4 + 2798848*y^2 + 107648, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648)
 

\( x^{28} + 116 x^{26} + 5162 x^{24} + 117160 x^{22} + 1511944 x^{20} + 11760544 x^{18} + 56876888 x^{16} + \cdots + 107648 \) Copy content Toggle raw display

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

Invariants

Degree:  $28$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(15909539583539461596949590009846202679084771957405055149146112\) \(\medspace = 2^{77}\cdot 29^{26}\) 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:  \(153.38\)
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:  $2^{11/4}29^{13/14}\approx 153.3816785195779$
Ramified primes:   \(2\), \(29\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(464=2^{4}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{464}(1,·)$, $\chi_{464}(323,·)$, $\chi_{464}(51,·)$, $\chi_{464}(257,·)$, $\chi_{464}(393,·)$, $\chi_{464}(267,·)$, $\chi_{464}(115,·)$, $\chi_{464}(401,·)$, $\chi_{464}(67,·)$, $\chi_{464}(25,·)$, $\chi_{464}(281,·)$, $\chi_{464}(283,·)$, $\chi_{464}(297,·)$, $\chi_{464}(161,·)$, $\chi_{464}(35,·)$, $\chi_{464}(65,·)$, $\chi_{464}(81,·)$, $\chi_{464}(169,·)$, $\chi_{464}(299,·)$, $\chi_{464}(419,·)$, $\chi_{464}(49,·)$, $\chi_{464}(91,·)$, $\chi_{464}(179,·)$, $\chi_{464}(233,·)$, $\chi_{464}(313,·)$, $\chi_{464}(347,·)$, $\chi_{464}(187,·)$, $\chi_{464}(411,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{8192}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{3944}a^{14}+\frac{1}{136}a^{12}+\frac{3}{34}a^{10}+\frac{1}{17}a^{8}+\frac{7}{34}a^{6}-\frac{1}{34}a^{4}-\frac{6}{17}a^{2}-\frac{4}{17}$, $\frac{1}{3944}a^{15}+\frac{1}{136}a^{13}+\frac{3}{34}a^{11}+\frac{1}{17}a^{9}+\frac{7}{34}a^{7}-\frac{1}{34}a^{5}-\frac{6}{17}a^{3}-\frac{4}{17}a$, $\frac{1}{7888}a^{16}+\frac{7}{17}$, $\frac{1}{7888}a^{17}+\frac{7}{17}a$, $\frac{1}{7888}a^{18}+\frac{7}{17}a^{2}$, $\frac{1}{7888}a^{19}+\frac{7}{17}a^{3}$, $\frac{1}{646816}a^{20}-\frac{9}{161704}a^{18}-\frac{1}{40426}a^{16}+\frac{1}{40426}a^{14}+\frac{21}{5576}a^{12}+\frac{143}{2788}a^{10}+\frac{339}{2788}a^{8}-\frac{261}{1394}a^{6}+\frac{10}{697}a^{4}-\frac{167}{697}a^{2}-\frac{55}{697}$, $\frac{1}{646816}a^{21}-\frac{9}{161704}a^{19}-\frac{1}{40426}a^{17}+\frac{1}{40426}a^{15}+\frac{21}{5576}a^{13}+\frac{143}{2788}a^{11}+\frac{339}{2788}a^{9}-\frac{261}{1394}a^{7}+\frac{10}{697}a^{5}-\frac{167}{697}a^{3}-\frac{55}{697}a$, $\frac{1}{646816}a^{22}+\frac{7}{323408}a^{16}+\frac{15}{161704}a^{14}+\frac{38}{697}a^{12}-\frac{335}{2788}a^{10}-\frac{331}{2788}a^{8}+\frac{95}{1394}a^{6}-\frac{135}{697}a^{4}+\frac{165}{697}a^{2}+\frac{193}{697}$, $\frac{1}{646816}a^{23}+\frac{7}{323408}a^{17}+\frac{15}{161704}a^{15}+\frac{38}{697}a^{13}-\frac{335}{2788}a^{11}-\frac{331}{2788}a^{9}+\frac{95}{1394}a^{7}-\frac{135}{697}a^{5}+\frac{165}{697}a^{3}+\frac{193}{697}a$, $\frac{1}{1297512896}a^{24}-\frac{209}{648756448}a^{22}-\frac{3}{11185456}a^{20}-\frac{346}{20273639}a^{18}-\frac{15513}{324378224}a^{16}-\frac{1883}{162189112}a^{14}+\frac{70603}{2796364}a^{12}-\frac{212697}{2796364}a^{10}+\frac{83167}{1398182}a^{8}+\frac{155351}{699091}a^{6}-\frac{59686}{699091}a^{4}-\frac{226218}{699091}a^{2}+\frac{340652}{699091}$, $\frac{1}{1297512896}a^{25}-\frac{209}{648756448}a^{23}-\frac{3}{11185456}a^{21}-\frac{346}{20273639}a^{19}-\frac{15513}{324378224}a^{17}-\frac{1883}{162189112}a^{15}+\frac{70603}{2796364}a^{13}-\frac{212697}{2796364}a^{11}+\frac{83167}{1398182}a^{9}+\frac{155351}{699091}a^{7}-\frac{59686}{699091}a^{5}-\frac{226218}{699091}a^{3}+\frac{340652}{699091}a$, $\frac{1}{57\!\cdots\!44}a^{26}+\frac{1728965}{57\!\cdots\!44}a^{24}-\frac{86574517}{14\!\cdots\!36}a^{22}-\frac{274957611}{358684383875684}a^{20}-\frac{74076924143}{14\!\cdots\!36}a^{18}-\frac{66088236903}{14\!\cdots\!36}a^{16}+\frac{177733425}{17496799213448}a^{14}+\frac{382677974351}{24736854060392}a^{12}-\frac{42916795289}{3092106757549}a^{10}+\frac{1241386027069}{12368427030196}a^{8}-\frac{990662959559}{6184213515098}a^{6}+\frac{149424870190}{3092106757549}a^{4}+\frac{522738787187}{3092106757549}a^{2}-\frac{1447807015172}{3092106757549}$, $\frac{1}{57\!\cdots\!44}a^{27}+\frac{1728965}{57\!\cdots\!44}a^{25}-\frac{86574517}{14\!\cdots\!36}a^{23}-\frac{274957611}{358684383875684}a^{21}-\frac{74076924143}{14\!\cdots\!36}a^{19}-\frac{66088236903}{14\!\cdots\!36}a^{17}+\frac{177733425}{17496799213448}a^{15}+\frac{382677974351}{24736854060392}a^{13}-\frac{42916795289}{3092106757549}a^{11}+\frac{1241386027069}{12368427030196}a^{9}-\frac{990662959559}{6184213515098}a^{7}+\frac{149424870190}{3092106757549}a^{5}+\frac{522738787187}{3092106757549}a^{3}-\frac{1447807015172}{3092106757549}a$ 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

not computed

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:  $13$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648)
 
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^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648, 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^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648);
 
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^28 + 116*x^26 + 5162*x^24 + 117160*x^22 + 1511944*x^20 + 11760544*x^18 + 56876888*x^16 + 173227904*x^14 + 331475104*x^12 + 392753728*x^10 + 281553344*x^8 + 117551616*x^6 + 26535232*x^4 + 2798848*x^2 + 107648);
 
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

$C_{28}$ (as 28T1):

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 28
The 28 conjugacy class representatives for $C_{28}$
Character table for $C_{28}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.1722368.6, 7.7.594823321.1, 14.14.742003380228915810271232.1

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 R $28$ $28$ ${\href{/padicField/7.7.0.1}{7} }^{4}$ $28$ $28$ ${\href{/padicField/17.2.0.1}{2} }^{14}$ $28$ ${\href{/padicField/23.7.0.1}{7} }^{4}$ R ${\href{/padicField/31.7.0.1}{7} }^{4}$ $28$ ${\href{/padicField/41.1.0.1}{1} }^{28}$ $28$ ${\href{/padicField/47.7.0.1}{7} }^{4}$ $28$ ${\href{/padicField/59.4.0.1}{4} }^{7}$

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 Deg $28$$4$$7$$77$
\(29\) Copy content Toggle raw display Deg $28$$14$$2$$26$