Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^28 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384)
gp: K = bnfinit(y^28 - 54*y^26 + 1300*y^24 - 18400*y^22 + 170016*y^20 - 1076768*y^18 + 4775232*y^16 - 14883840*y^14 + 32248320*y^12 - 47297536*y^10 + 44808192*y^8 - 25346048*y^6 + 7454720*y^4 - 860160*y^2 + 16384, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^28 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^28 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384)
\( x^{28} - 54 x^{26} + 1300 x^{24} - 18400 x^{22} + 170016 x^{20} - 1076768 x^{18} + 4775232 x^{16} + \cdots + 16384 \)
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: |
\(463028542684026225381227850734902390950731116969984\)
\(\medspace = 2^{42}\cdot 29^{26}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.49\) | 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: | $2^{3/2}29^{13/14}\approx 64.48905181635536$ | ||
| Ramified primes: |
\(2\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $28$ | 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$$\chi_{232}(129,·)$, $\chi_{232}(5,·)$, $\chi_{232}(1,·)$, $\chi_{232}(9,·)$, $\chi_{232}(109,·)$, $\chi_{232}(13,·)$, $\chi_{232}(45,·)$, $\chi_{232}(141,·)$, $\chi_{232}(209,·)$, $\chi_{232}(149,·)$, $\chi_{232}(121,·)$, $\chi_{232}(25,·)$, $\chi_{232}(93,·)$, $\chi_{232}(197,·)$, $\chi_{232}(161,·)$, $\chi_{232}(165,·)$, $\chi_{232}(65,·)$, $\chi_{232}(81,·)$, $\chi_{232}(169,·)$, $\chi_{232}(225,·)$, $\chi_{232}(173,·)$, $\chi_{232}(49,·)$, $\chi_{232}(181,·)$, $\chi_{232}(57,·)$, $\chi_{232}(33,·)$, $\chi_{232}(117,·)$, $\chi_{232}(125,·)$, $\chi_{232}(53,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{4096}a^{24}$, $\frac{1}{4096}a^{25}$, $\frac{1}{8192}a^{26}$, $\frac{1}{8192}a^{27}$
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 $2$)
| 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: |
$\frac{1}{8192}a^{26}-\frac{25}{4096}a^{24}+\frac{69}{512}a^{22}-\frac{443}{256}a^{20}+\frac{3667}{256}a^{18}-\frac{10251}{128}a^{16}+\frac{39441}{128}a^{14}-\frac{52221}{64}a^{12}+\frac{46827}{32}a^{10}-\frac{27479}{16}a^{8}+\frac{9941}{8}a^{6}-\frac{2001}{4}a^{4}+\frac{185}{2}a^{2}-5$, $\frac{1}{8}a^{6}-\frac{3}{2}a^{4}+\frac{9}{2}a^{2}-2$, $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-14a^{6}+\frac{105}{4}a^{4}-18a^{2}+2$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{63}{256}a^{20}-\frac{95}{32}a^{18}+\frac{2907}{128}a^{16}-\frac{459}{4}a^{14}+\frac{1547}{4}a^{12}-858a^{10}+\frac{19305}{16}a^{8}-1001a^{6}+429a^{4}-72a^{2}+2$, $\frac{1}{8}a^{6}-\frac{3}{2}a^{4}+\frac{9}{2}a^{2}-1$, $\frac{1}{256}a^{16}-\frac{1}{8}a^{14}+\frac{13}{8}a^{12}-11a^{10}+\frac{165}{4}a^{8}-84a^{6}+84a^{4}-32a^{2}+2$, $\frac{1}{4096}a^{24}-\frac{3}{256}a^{22}+\frac{63}{256}a^{20}-\frac{95}{32}a^{18}+\frac{2907}{128}a^{16}-\frac{459}{4}a^{14}+\frac{1547}{4}a^{12}-858a^{10}+\frac{9653}{8}a^{8}-1002a^{6}+434a^{4}-80a^{2}+4$, $\frac{1}{16}a^{8}-a^{6}+5a^{4}-8a^{2}+3$, $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-14a^{6}+\frac{105}{4}a^{4}-18a^{2}+3$, $\frac{1}{8192}a^{26}-\frac{25}{4096}a^{24}+\frac{69}{512}a^{22}-\frac{1771}{1024}a^{20}+\frac{7315}{512}a^{18}-\frac{20349}{256}a^{16}+\frac{38759}{128}a^{14}-\frac{25187}{32}a^{12}+\frac{43681}{32}a^{10}-\frac{6025}{4}a^{8}+\frac{3857}{4}a^{6}-\frac{1169}{4}a^{4}+21a^{2}$, $\frac{1}{64}a^{12}-\frac{3}{8}a^{10}+\frac{27}{8}a^{8}-\frac{111}{8}a^{6}+\frac{99}{4}a^{4}-\frac{27}{2}a^{2}+1$, $\frac{1}{4}a^{4}-2a^{2}+3$, $\frac{1}{2}a^{2}-1$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}+\frac{8855}{512}a^{19}-\frac{13167}{128}a^{17}+\frac{6783}{16}a^{15}-\frac{4845}{4}a^{13}+\frac{37791}{16}a^{11}-\frac{12155}{4}a^{9}+2431a^{7}-1092a^{5}+\frac{455}{2}a^{3}-14a+1$, $\frac{1}{8192}a^{27}-\frac{1}{8192}a^{26}-\frac{13}{2048}a^{25}+\frac{25}{4096}a^{24}+\frac{75}{512}a^{23}-\frac{69}{512}a^{22}-\frac{253}{128}a^{21}+\frac{1771}{1024}a^{20}+\frac{8855}{512}a^{19}-\frac{7315}{512}a^{18}-\frac{13167}{128}a^{17}+\frac{20349}{256}a^{16}+\frac{6783}{16}a^{15}-\frac{4845}{16}a^{14}-\frac{4845}{4}a^{13}+\frac{12597}{16}a^{12}+\frac{37791}{16}a^{11}-\frac{21879}{16}a^{10}-\frac{12155}{4}a^{9}+\frac{12155}{8}a^{8}+2431a^{7}-1001a^{6}-1092a^{5}+\frac{1365}{4}a^{4}+\frac{455}{2}a^{3}-\frac{91}{2}a^{2}-14a+1$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}+\frac{8855}{512}a^{19}-\frac{13167}{128}a^{17}+\frac{6783}{16}a^{15}-\frac{4845}{4}a^{13}-\frac{1}{64}a^{12}+\frac{37791}{16}a^{11}+\frac{3}{8}a^{10}-\frac{12155}{4}a^{9}-\frac{27}{8}a^{8}+2431a^{7}+14a^{6}-1092a^{5}-\frac{105}{4}a^{4}+\frac{455}{2}a^{3}+18a^{2}-14a-2$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}+\frac{8855}{512}a^{19}-\frac{13167}{128}a^{17}+\frac{6783}{16}a^{15}-\frac{4845}{4}a^{13}+\frac{37791}{16}a^{11}-\frac{12155}{4}a^{9}+2431a^{7}-1092a^{5}+\frac{455}{2}a^{3}-\frac{1}{2}a^{2}-14a+2$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}-\frac{1}{1024}a^{20}+\frac{8855}{512}a^{19}+\frac{5}{128}a^{18}-\frac{13167}{128}a^{17}-\frac{85}{128}a^{16}+\frac{6783}{16}a^{15}+\frac{25}{4}a^{14}-\frac{4845}{4}a^{13}-\frac{2275}{64}a^{12}+\frac{37791}{16}a^{11}+\frac{1001}{8}a^{10}-\frac{12155}{4}a^{9}-\frac{2145}{8}a^{8}+2431a^{7}+330a^{6}-1092a^{5}-\frac{825}{4}a^{4}+\frac{455}{2}a^{3}+50a^{2}-14a-2$, $\frac{1}{8192}a^{27}+\frac{1}{8192}a^{26}-\frac{13}{2048}a^{25}-\frac{13}{2048}a^{24}+\frac{75}{512}a^{23}+\frac{299}{2048}a^{22}-\frac{253}{128}a^{21}-\frac{1001}{512}a^{20}+\frac{8855}{512}a^{19}+\frac{8645}{512}a^{18}-\frac{13167}{128}a^{17}-\frac{12597}{128}a^{16}+\frac{6783}{16}a^{15}+\frac{12597}{32}a^{14}-\frac{4845}{4}a^{13}-\frac{8619}{8}a^{12}+\frac{37791}{16}a^{11}+\frac{31603}{16}a^{10}-\frac{12155}{4}a^{9}-\frac{9295}{4}a^{8}+2431a^{7}+\frac{13013}{8}a^{6}-1092a^{5}-\frac{1183}{2}a^{4}+\frac{455}{2}a^{3}+\frac{169}{2}a^{2}-14a-2$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}+\frac{8855}{512}a^{19}-\frac{13167}{128}a^{17}+\frac{6783}{16}a^{15}-\frac{4845}{4}a^{13}+\frac{37791}{16}a^{11}-\frac{12155}{4}a^{9}+2431a^{7}-1092a^{5}+\frac{1}{4}a^{4}+\frac{455}{2}a^{3}-2a^{2}-14a+2$, $\frac{1}{4096}a^{25}+\frac{1}{4096}a^{24}-\frac{25}{2048}a^{23}-\frac{3}{256}a^{22}+\frac{275}{1024}a^{21}+\frac{251}{1024}a^{20}-\frac{875}{256}a^{19}-\frac{375}{128}a^{18}+\frac{7125}{256}a^{17}+\frac{1411}{64}a^{16}-\frac{4845}{32}a^{15}-\frac{13889}{128}a^{14}+\frac{8925}{16}a^{13}+\frac{11245}{32}a^{12}-\frac{5525}{4}a^{11}-\frac{23517}{32}a^{10}+\frac{17875}{8}a^{9}+\frac{15171}{16}a^{8}-\frac{8937}{4}a^{7}-\frac{5551}{8}a^{6}+\frac{2499}{2}a^{5}+247a^{4}-318a^{3}-33a^{2}+19a+2$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}+\frac{1}{2048}a^{22}-\frac{253}{128}a^{21}-\frac{11}{512}a^{20}+\frac{8855}{512}a^{19}+\frac{209}{512}a^{18}-\frac{13167}{128}a^{17}-\frac{561}{128}a^{16}+\frac{6783}{16}a^{15}+\frac{935}{32}a^{14}-\frac{4845}{4}a^{13}-\frac{1001}{8}a^{12}+\frac{37791}{16}a^{11}+\frac{11011}{32}a^{10}-\frac{12155}{4}a^{9}-\frac{4719}{8}a^{8}+2431a^{7}+\frac{4719}{8}a^{6}-1092a^{5}-\frac{605}{2}a^{4}+\frac{455}{2}a^{3}+\frac{121}{2}a^{2}-14a-2$, $\frac{1}{8192}a^{26}-\frac{25}{4096}a^{24}+\frac{1}{2048}a^{23}+\frac{69}{512}a^{22}-\frac{23}{1024}a^{21}-\frac{1771}{1024}a^{20}+\frac{115}{256}a^{19}+\frac{7315}{512}a^{18}-\frac{1311}{256}a^{17}-\frac{10175}{128}a^{16}+\frac{1173}{32}a^{15}+\frac{4847}{16}a^{14}-\frac{2737}{16}a^{13}-\frac{50493}{64}a^{12}+\frac{2093}{4}a^{11}+\frac{22061}{16}a^{10}-\frac{4111}{4}a^{9}-1564a^{8}+\frac{4929}{4}a^{7}+1099a^{6}-\frac{3261}{4}a^{5}-\frac{1805}{4}a^{4}+\frac{471}{2}a^{3}+93a^{2}-9a-1$, $\frac{1}{8192}a^{27}-\frac{1}{8192}a^{26}-\frac{25}{4096}a^{25}+\frac{23}{4096}a^{24}+\frac{69}{512}a^{23}-\frac{115}{1024}a^{22}-\frac{1771}{1024}a^{21}+\frac{655}{512}a^{20}+\frac{3657}{256}a^{19}-\frac{2337}{256}a^{18}-\frac{20331}{256}a^{17}+\frac{10813}{256}a^{16}+\frac{1207}{4}a^{15}-\frac{16199}{128}a^{14}-\frac{12457}{16}a^{13}+\frac{15171}{64}a^{12}+\frac{42393}{32}a^{11}-\frac{8151}{32}a^{10}-\frac{22307}{16}a^{9}+\frac{2067}{16}a^{8}+\frac{1571}{2}a^{7}-\frac{85}{4}a^{6}-138a^{5}+\frac{59}{4}a^{4}-\frac{95}{2}a^{3}-12a^{2}+15a+2$, $\frac{1}{8192}a^{26}-\frac{13}{2048}a^{24}+\frac{299}{2048}a^{22}-\frac{1001}{512}a^{20}+\frac{8645}{512}a^{18}-\frac{12597}{128}a^{16}+\frac{12597}{32}a^{14}+\frac{1}{64}a^{13}-\frac{8619}{8}a^{12}-\frac{13}{32}a^{11}+\frac{31603}{16}a^{10}+\frac{65}{16}a^{9}-\frac{9295}{4}a^{8}-\frac{39}{2}a^{7}+\frac{13013}{8}a^{6}+\frac{91}{2}a^{5}-\frac{1183}{2}a^{4}-\frac{91}{2}a^{3}+\frac{169}{2}a^{2}+13a-1$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}+\frac{8855}{512}a^{19}-\frac{13167}{128}a^{17}+\frac{6783}{16}a^{15}-\frac{4845}{4}a^{13}+\frac{37791}{16}a^{11}-\frac{1}{32}a^{10}-\frac{12155}{4}a^{9}+\frac{5}{8}a^{8}+2431a^{7}-\frac{35}{8}a^{6}-1092a^{5}+\frac{25}{2}a^{4}+\frac{455}{2}a^{3}-\frac{25}{2}a^{2}-14a+2$, $\frac{1}{8192}a^{27}-\frac{13}{2048}a^{25}+\frac{75}{512}a^{23}-\frac{253}{128}a^{21}+\frac{8855}{512}a^{19}-\frac{1}{512}a^{18}-\frac{13167}{128}a^{17}+\frac{9}{128}a^{16}+\frac{6783}{16}a^{15}-\frac{135}{128}a^{14}-\frac{4845}{4}a^{13}+\frac{273}{32}a^{12}+\frac{37791}{16}a^{11}-\frac{1287}{32}a^{10}-\frac{12155}{4}a^{9}+\frac{891}{8}a^{8}+2431a^{7}-\frac{693}{4}a^{6}-1092a^{5}+135a^{4}+\frac{455}{2}a^{3}-\frac{81}{2}a^{2}-14a+2$
| 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: | \( 18257396114183336 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{28}\cdot(2\pi)^{0}\cdot 18257396114183336 \cdot 1}{2\cdot\sqrt{463028542684026225381227850734902390950731116969984}}\cr\approx \mathstrut & 0.113879313246372 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^28 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384)
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 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384, 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 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384);
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 - 54*x^26 + 1300*x^24 - 18400*x^22 + 170016*x^20 - 1076768*x^18 + 4775232*x^16 - 14883840*x^14 + 32248320*x^12 - 47297536*x^10 + 44808192*x^8 - 25346048*x^6 + 7454720*x^4 - 860160*x^2 + 16384);
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
$C_2\times C_{14}$ (as 28T2):
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)
| An abelian group of order 28 |
| The 28 conjugacy class representatives for $C_2\times C_{14}$ |
| Character table for $C_2\times C_{14}$ |
Intermediate fields
| \(\Q(\sqrt{58}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{2}, \sqrt{29})\), 7.7.594823321.1, 14.14.21518098026638558497865728.1, 14.14.742003380228915810271232.1, \(\Q(\zeta_{29})^+\) |
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 | ${\href{/padicField/3.14.0.1}{14} }^{2}$ | ${\href{/padicField/5.14.0.1}{14} }^{2}$ | ${\href{/padicField/7.7.0.1}{7} }^{4}$ | ${\href{/padicField/11.14.0.1}{14} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{14}$ | ${\href{/padicField/19.14.0.1}{14} }^{2}$ | ${\href{/padicField/23.7.0.1}{7} }^{4}$ | R | ${\href{/padicField/31.14.0.1}{14} }^{2}$ | ${\href{/padicField/37.14.0.1}{14} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{14}$ | ${\href{/padicField/43.14.0.1}{14} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }^{2}$ | ${\href{/padicField/53.14.0.1}{14} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{14}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $28$ | $2$ | $14$ | $42$ | |||
|
\(29\)
| Deg $28$ | $14$ | $2$ | $26$ |