Properties

Label 28.28.134...536.1
Degree $28$
Signature $[28, 0]$
Discriminant $1.343\times 10^{52}$
Root discriminant \(72.73\)
Ramified primes $2,29$
Class number $1$ (GRH)
Class group trivial (GRH)
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 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136)
 
gp: K = bnfinit(y^28 - 58*y^26 + 1508*y^24 - 23200*y^22 + 234784*y^20 - 1643488*y^18 + 8145984*y^16 - 28775424*y^14 + 71938560*y^12 - 124693504*y^10 + 144381952*y^8 - 105005056*y^6 + 43237376*y^4 - 8314880*y^2 + 475136, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136)
 

\( x^{28} - 58 x^{26} + 1508 x^{24} - 23200 x^{22} + 234784 x^{20} - 1643488 x^{18} + 8145984 x^{16} - 28775424 x^{14} + 71938560 x^{12} - 124693504 x^{10} + \cdots + 475136 \) 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:  $[28, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(13427827737836760536055607671312169337571202392129536\) \(\medspace = 2^{42}\cdot 29^{27}\) 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:  \(72.73\)
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:  $2^{3/2}29^{27/28}\approx 72.7301513733453$
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:  \(\Q(\sqrt{29}) \)
$\card{ \Gal(K/\Q) }$:  $28$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(232=2^{3}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{232}(1,·)$, $\chi_{232}(3,·)$, $\chi_{232}(147,·)$, $\chi_{232}(81,·)$, $\chi_{232}(129,·)$, $\chi_{232}(9,·)$, $\chi_{232}(11,·)$, $\chi_{232}(161,·)$, $\chi_{232}(43,·)$, $\chi_{232}(75,·)$, $\chi_{232}(19,·)$, $\chi_{232}(121,·)$, $\chi_{232}(225,·)$, $\chi_{232}(25,·)$, $\chi_{232}(155,·)$, $\chi_{232}(33,·)$, $\chi_{232}(163,·)$, $\chi_{232}(131,·)$, $\chi_{232}(65,·)$, $\chi_{232}(209,·)$, $\chi_{232}(169,·)$, $\chi_{232}(171,·)$, $\chi_{232}(27,·)$, $\chi_{232}(49,·)$, $\chi_{232}(211,·)$, $\chi_{232}(99,·)$, $\chi_{232}(57,·)$, $\chi_{232}(195,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$ 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 $1$ (assuming GRH)

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:  $27$
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:   $\frac{1}{512}a^{18}-\frac{9}{128}a^{16}+\frac{135}{128}a^{14}-\frac{273}{32}a^{12}+\frac{1287}{32}a^{10}-\frac{891}{8}a^{8}+\frac{693}{4}a^{6}-135a^{4}+\frac{81}{2}a^{2}-2$, $\frac{1}{2048}a^{22}-\frac{11}{512}a^{20}+\frac{209}{512}a^{18}-\frac{561}{128}a^{16}+\frac{935}{32}a^{14}-\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}-\frac{4719}{8}a^{8}+\frac{4719}{8}a^{6}-\frac{605}{2}a^{4}+\frac{121}{2}a^{2}-2$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{147}{4}a^{6}-49a^{4}+\frac{49}{2}a^{2}-2$, $\frac{1}{1024}a^{20}-\frac{5}{128}a^{18}+\frac{85}{128}a^{16}-\frac{25}{4}a^{14}+\frac{2275}{64}a^{12}-\frac{1001}{8}a^{10}+\frac{2145}{8}a^{8}-330a^{6}+\frac{825}{4}a^{4}-50a^{2}+2$, $\frac{1}{2048}a^{22}-\frac{11}{512}a^{20}+\frac{209}{512}a^{18}-\frac{561}{128}a^{16}+\frac{3741}{128}a^{14}-\frac{4011}{32}a^{12}+\frac{693}{2}a^{10}-603a^{8}+\frac{5013}{8}a^{6}-\frac{703}{2}a^{4}+85a^{2}-5$, $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-14a^{6}+\frac{105}{4}a^{4}-18a^{2}+3$, $\frac{1}{8192}a^{26}-\frac{27}{4096}a^{24}+\frac{81}{512}a^{22}-\frac{2277}{1024}a^{20}+\frac{10395}{512}a^{18}-\frac{32319}{256}a^{16}+\frac{8721}{16}a^{14}-\frac{26163}{16}a^{12}+\frac{53703}{16}a^{10}-\frac{36465}{8}a^{8}+3861a^{6}-\frac{7371}{4}a^{4}+\frac{819}{2}a^{2}-27$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{63}{256}a^{20}-\frac{95}{32}a^{18}+\frac{2907}{128}a^{16}-\frac{459}{4}a^{14}+\frac{1547}{4}a^{12}-858a^{10}+\frac{19305}{16}a^{8}-1001a^{6}+429a^{4}-72a^{2}+3$, $\frac{1}{2048}a^{22}-\frac{11}{512}a^{20}+\frac{209}{512}a^{18}-\frac{561}{128}a^{16}+\frac{935}{32}a^{14}-\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}-\frac{4719}{8}a^{8}+\frac{4719}{8}a^{6}-\frac{605}{2}a^{4}+\frac{121}{2}a^{2}-3$, $\frac{1}{8192}a^{26}-\frac{27}{4096}a^{24}+\frac{81}{512}a^{22}-\frac{2277}{1024}a^{20}+\frac{10395}{512}a^{18}-\frac{16159}{128}a^{16}+\frac{69751}{128}a^{14}-\frac{52267}{32}a^{12}+\frac{106977}{32}a^{10}-\frac{18015}{4}a^{8}+\frac{14961}{4}a^{6}-\frac{6839}{4}a^{4}+\frac{705}{2}a^{2}-20$, $\frac{1}{8192}a^{26}-\frac{27}{4096}a^{24}+\frac{81}{512}a^{22}-\frac{2277}{1024}a^{20}+\frac{10395}{512}a^{18}-\frac{32319}{256}a^{16}+\frac{8721}{16}a^{14}-\frac{26163}{16}a^{12}+\frac{107405}{32}a^{10}-\frac{9115}{2}a^{8}+\frac{30853}{8}a^{6}-\frac{7321}{4}a^{4}+397a^{2}-25$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{147}{4}a^{6}-49a^{4}+\frac{49}{2}a^{2}-3$, $\frac{1}{2048}a^{22}-\frac{11}{512}a^{20}+\frac{209}{512}a^{18}-\frac{561}{128}a^{16}+\frac{935}{32}a^{14}-\frac{1001}{8}a^{12}+\frac{11011}{32}a^{10}-\frac{9439}{16}a^{8}+591a^{6}-309a^{4}+73a^{2}-6$, $\frac{1}{1024}a^{21}-\frac{21}{512}a^{19}+\frac{189}{256}a^{17}-\frac{1}{256}a^{16}-\frac{119}{16}a^{15}+\frac{1}{8}a^{14}+\frac{735}{16}a^{13}-\frac{13}{8}a^{12}-\frac{5733}{32}a^{11}+11a^{10}+\frac{7007}{16}a^{9}-\frac{165}{4}a^{8}-\frac{1287}{2}a^{7}+84a^{6}+\frac{2079}{4}a^{5}-84a^{4}-\frac{385}{2}a^{3}+32a^{2}+21a-1$, $\frac{1}{128}a^{14}-\frac{7}{32}a^{12}+\frac{77}{32}a^{10}-\frac{105}{8}a^{8}+\frac{1}{8}a^{7}+\frac{147}{4}a^{6}-\frac{7}{4}a^{5}-49a^{4}+7a^{3}+\frac{49}{2}a^{2}-7a-1$, $\frac{1}{2048}a^{23}-\frac{11}{512}a^{21}+\frac{209}{512}a^{19}-\frac{561}{128}a^{17}+\frac{935}{32}a^{15}-\frac{1}{128}a^{14}-\frac{1001}{8}a^{13}+\frac{7}{32}a^{12}+\frac{11011}{32}a^{11}-\frac{77}{32}a^{10}-\frac{4719}{8}a^{9}+\frac{105}{8}a^{8}+\frac{4719}{8}a^{7}-\frac{147}{4}a^{6}-\frac{605}{2}a^{5}+49a^{4}+\frac{121}{2}a^{3}-24a^{2}-2a+2$, $\frac{1}{8192}a^{26}-\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}-\frac{1}{1024}a^{21}-\frac{1001}{512}a^{20}+\frac{21}{512}a^{19}+\frac{8645}{512}a^{18}-\frac{189}{256}a^{17}-\frac{12597}{128}a^{16}+\frac{119}{16}a^{15}+\frac{12597}{32}a^{14}-\frac{735}{16}a^{13}-\frac{8619}{8}a^{12}+\frac{1433}{8}a^{11}+\frac{63205}{32}a^{10}-\frac{1749}{4}a^{9}-\frac{18585}{8}a^{8}+638a^{7}+\frac{6489}{4}a^{6}-\frac{1001}{2}a^{5}-579a^{4}+165a^{3}+72a^{2}-10a-2$, $\frac{1}{8192}a^{27}-\frac{27}{4096}a^{25}+\frac{81}{512}a^{23}-\frac{569}{256}a^{21}+\frac{5187}{256}a^{19}-\frac{16065}{128}a^{17}+\frac{4301}{8}a^{15}-\frac{6357}{4}a^{13}+\frac{101673}{32}a^{11}+\frac{1}{32}a^{10}-\frac{65923}{16}a^{9}-\frac{5}{8}a^{8}+\frac{6435}{2}a^{7}+\frac{17}{4}a^{6}-1323a^{5}-11a^{4}+217a^{3}+8a^{2}-6a-2$, $\frac{1}{8192}a^{26}-\frac{27}{4096}a^{24}-\frac{1}{2048}a^{23}+\frac{81}{512}a^{22}+\frac{23}{1024}a^{21}-\frac{2277}{1024}a^{20}-\frac{115}{256}a^{19}+\frac{2599}{128}a^{18}+\frac{1311}{256}a^{17}-\frac{32337}{256}a^{16}-\frac{1173}{32}a^{15}+\frac{69903}{128}a^{14}+\frac{2737}{16}a^{13}-\frac{52599}{32}a^{12}-\frac{2093}{4}a^{11}+\frac{108693}{32}a^{10}+\frac{16445}{16}a^{9}-\frac{9339}{2}a^{8}-\frac{9867}{8}a^{7}+\frac{16137}{4}a^{6}+\frac{1645}{2}a^{5}-\frac{7911}{4}a^{4}-\frac{511}{2}a^{3}+450a^{2}+28a-31$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{63}{256}a^{20}+\frac{1}{512}a^{19}-\frac{95}{32}a^{18}-\frac{19}{256}a^{17}+\frac{2907}{128}a^{16}+\frac{19}{16}a^{15}-\frac{14687}{128}a^{14}-\frac{665}{64}a^{13}+\frac{12369}{32}a^{12}+\frac{1729}{32}a^{11}-\frac{27379}{32}a^{10}-\frac{2717}{16}a^{9}+\frac{19095}{16}a^{8}+\frac{627}{2}a^{7}-\frac{3857}{4}a^{6}-\frac{1255}{4}a^{5}+380a^{4}+145a^{3}-\frac{95}{2}a^{2}-24a-2$, $\frac{1}{4}a^{4}+\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-3a-1$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{1}{1024}a^{21}+\frac{63}{256}a^{20}-\frac{21}{512}a^{19}-\frac{1519}{512}a^{18}+\frac{189}{256}a^{17}+\frac{1449}{64}a^{16}-\frac{119}{16}a^{15}-\frac{14553}{128}a^{14}+\frac{735}{16}a^{13}+\frac{12103}{32}a^{12}-\frac{5733}{32}a^{11}-\frac{26169}{32}a^{10}+\frac{7007}{16}a^{9}+\frac{17523}{16}a^{8}-\frac{1287}{2}a^{7}-\frac{3311}{4}a^{6}+\frac{2079}{4}a^{5}+294a^{4}-192a^{3}-\frac{63}{2}a^{2}+18a+2$, $\frac{1}{8192}a^{26}-\frac{27}{4096}a^{24}+\frac{81}{512}a^{22}+\frac{1}{1024}a^{21}-\frac{2277}{1024}a^{20}-\frac{21}{512}a^{19}+\frac{10395}{512}a^{18}+\frac{189}{256}a^{17}-\frac{32319}{256}a^{16}-\frac{119}{16}a^{15}+\frac{8721}{16}a^{14}+\frac{735}{16}a^{13}-\frac{104653}{64}a^{12}-\frac{5733}{32}a^{11}+\frac{53709}{16}a^{10}+\frac{3503}{8}a^{9}-\frac{9123}{2}a^{8}-\frac{5139}{8}a^{7}+3875a^{6}+513a^{5}-1869a^{4}-\frac{355}{2}a^{3}+\frac{855}{2}a^{2}+12a-27$, $\frac{1}{8192}a^{26}-\frac{1}{4096}a^{25}-\frac{27}{4096}a^{24}+\frac{25}{2048}a^{23}+\frac{81}{512}a^{22}-\frac{275}{1024}a^{21}-\frac{1139}{512}a^{20}+\frac{875}{256}a^{19}+\frac{10415}{512}a^{18}-\frac{7125}{256}a^{17}-\frac{32489}{256}a^{16}+\frac{4845}{32}a^{15}+\frac{8821}{16}a^{14}-\frac{8925}{16}a^{13}-\frac{106927}{64}a^{12}+\frac{5525}{4}a^{11}+\frac{55705}{16}a^{10}-\frac{17875}{8}a^{9}-\frac{19305}{4}a^{8}+\frac{17875}{8}a^{7}+4191a^{6}-\frac{2503}{2}a^{5}-2049a^{4}+\frac{655}{2}a^{3}+\frac{919}{2}a^{2}-30a-31$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}-\frac{1}{1024}a^{21}+\frac{63}{256}a^{20}+\frac{21}{512}a^{19}-\frac{95}{32}a^{18}-\frac{189}{256}a^{17}+\frac{2907}{128}a^{16}+\frac{119}{16}a^{15}-\frac{459}{4}a^{14}-\frac{2941}{64}a^{13}+\frac{1547}{4}a^{12}+\frac{2873}{16}a^{11}-858a^{10}-442a^{9}+\frac{2413}{2}a^{8}+663a^{7}-1000a^{6}-\frac{2261}{4}a^{5}+424a^{4}+238a^{3}-64a^{2}-34a-2$, $\frac{1}{1024}a^{20}-\frac{5}{128}a^{18}+\frac{85}{128}a^{16}-\frac{1}{128}a^{15}-\frac{25}{4}a^{14}+\frac{15}{64}a^{13}+\frac{2275}{64}a^{12}-\frac{45}{16}a^{11}-\frac{4003}{32}a^{10}+\frac{275}{16}a^{9}+\frac{535}{2}a^{8}-\frac{225}{4}a^{7}-\frac{2605}{8}a^{6}+\frac{377}{4}a^{5}+\frac{775}{4}a^{4}-\frac{135}{2}a^{3}-\frac{75}{2}a^{2}+10a+2$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{63}{256}a^{20}-\frac{95}{32}a^{18}+\frac{1}{256}a^{17}+\frac{2907}{128}a^{16}-\frac{17}{128}a^{15}-\frac{459}{4}a^{14}+\frac{119}{64}a^{13}+\frac{1547}{4}a^{12}-\frac{221}{16}a^{11}-858a^{10}+\frac{935}{16}a^{9}+\frac{19305}{16}a^{8}-\frac{561}{4}a^{7}-1001a^{6}+\frac{357}{2}a^{5}+429a^{4}-102a^{3}-72a^{2}+17a+1$ 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:  \( 170028857236944540 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 170028857236944540 \cdot 1}{2\cdot\sqrt{13427827737836760536055607671312169337571202392129536}}\cr\approx \mathstrut & 0.196938052216447 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^28 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136)
 
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 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136, 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 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136);
 
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 - 58*x^26 + 1508*x^24 - 23200*x^22 + 234784*x^20 - 1643488*x^18 + 8145984*x^16 - 28775424*x^14 + 71938560*x^12 - 124693504*x^10 + 144381952*x^8 - 105005056*x^6 + 43237376*x^4 - 8314880*x^2 + 475136);
 
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

\(\Q(\sqrt{29}) \), 4.4.1560896.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\)

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 $28$ ${\href{/padicField/5.7.0.1}{7} }^{4}$ ${\href{/padicField/7.14.0.1}{14} }^{2}$ $28$ ${\href{/padicField/13.7.0.1}{7} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{7}$ $28$ ${\href{/padicField/23.14.0.1}{14} }^{2}$ R $28$ $28$ ${\href{/padicField/41.4.0.1}{4} }^{7}$ $28$ $28$ ${\href{/padicField/53.14.0.1}{14} }^{2}$ ${\href{/padicField/59.1.0.1}{1} }^{28}$

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$$2$$14$$42$
\(29\) Copy content Toggle raw display Deg $28$$28$$1$$27$