Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80)
gp: K = bnfinit(y^16 - 4*y^15 - 10*y^14 + 127*y^13 - 380*y^12 - 84*y^11 + 890*y^10 + 591*y^9 - 2244*y^8 - 1586*y^7 + 3018*y^6 + 2649*y^5 - 501*y^4 - 2078*y^3 - 204*y^2 + 520*y - 80, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80)
\( x^{16} - 4 x^{15} - 10 x^{14} + 127 x^{13} - 380 x^{12} - 84 x^{11} + 890 x^{10} + 591 x^{9} - 2244 x^{8} + \cdots - 80 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[8, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(43149301773875176172265625\)
\(\medspace = 5^{8}\cdot 101^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(40.01\) | 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: | $5^{1/2}101^{3/4}\approx 71.24034803863643$ | ||
| Ramified primes: |
\(5\), \(101\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{10}a^{8}+\frac{1}{10}a^{7}-\frac{3}{10}a^{6}+\frac{3}{10}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{10}a^{2}$, $\frac{1}{10}a^{9}+\frac{1}{10}a^{7}+\frac{1}{10}a^{6}+\frac{3}{10}a^{4}+\frac{3}{10}a^{3}+\frac{1}{10}a^{2}-\frac{1}{2}a$, $\frac{1}{20}a^{10}-\frac{1}{20}a^{8}-\frac{1}{20}a^{7}+\frac{3}{10}a^{6}-\frac{3}{20}a^{5}-\frac{3}{20}a^{4}-\frac{1}{20}a^{3}-\frac{3}{20}a^{2}$, $\frac{1}{100}a^{11}+\frac{1}{50}a^{10}+\frac{3}{100}a^{9}+\frac{3}{100}a^{8}+\frac{6}{25}a^{7}+\frac{9}{20}a^{6}+\frac{39}{100}a^{5}-\frac{37}{100}a^{4}+\frac{43}{100}a^{3}-\frac{2}{25}a^{2}-\frac{3}{10}a+\frac{2}{5}$, $\frac{1}{100}a^{12}-\frac{1}{100}a^{10}-\frac{3}{100}a^{9}-\frac{1}{50}a^{8}-\frac{23}{100}a^{7}+\frac{9}{100}a^{6}+\frac{1}{4}a^{5}-\frac{43}{100}a^{4}-\frac{7}{50}a^{3}+\frac{3}{50}a^{2}+\frac{1}{5}$, $\frac{1}{1000}a^{13}-\frac{1}{1000}a^{12}+\frac{1}{250}a^{11}+\frac{1}{125}a^{10}+\frac{3}{500}a^{9}-\frac{3}{500}a^{8}+\frac{121}{500}a^{7}-\frac{69}{1000}a^{6}+\frac{327}{1000}a^{5}-\frac{193}{500}a^{4}+\frac{1}{200}a^{3}-\frac{32}{125}a^{2}+\frac{1}{50}a+\frac{12}{25}$, $\frac{1}{5000}a^{14}-\frac{1}{2500}a^{13}+\frac{1}{1000}a^{12}+\frac{3}{625}a^{11}-\frac{14}{625}a^{10}+\frac{37}{1250}a^{9}-\frac{121}{2500}a^{8}+\frac{219}{5000}a^{7}-\frac{113}{625}a^{6}-\frac{83}{5000}a^{5}+\frac{1301}{5000}a^{4}-\frac{151}{5000}a^{3}-\frac{817}{2500}a^{2}-\frac{57}{250}a-\frac{42}{125}$, $\frac{1}{74\!\cdots\!00}a^{15}-\frac{120130897589}{37\!\cdots\!00}a^{14}-\frac{678573252093}{74\!\cdots\!00}a^{13}-\frac{3705943374989}{18\!\cdots\!50}a^{12}-\frac{1084382995617}{928498903184375}a^{11}+\frac{13651493851121}{742799122547500}a^{10}+\frac{14634723379113}{371399561273750}a^{9}-\frac{226818551606789}{74\!\cdots\!00}a^{8}-\frac{606860798504949}{37\!\cdots\!00}a^{7}-\frac{773516039812879}{74\!\cdots\!00}a^{6}+\frac{196762202949809}{74\!\cdots\!00}a^{5}+\frac{21\!\cdots\!23}{74\!\cdots\!00}a^{4}+\frac{287714326333499}{928498903184375}a^{3}-\frac{55284628272157}{928498903184375}a^{2}-\frac{20258335610901}{185699780636875}a-\frac{40860712894583}{185699780636875}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $5$ |
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: | $11$ | 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: |
$\frac{128485382}{660853311875}a^{15}-\frac{843478143}{2643413247500}a^{14}-\frac{22823614197}{5286826495000}a^{13}+\frac{120201949199}{5286826495000}a^{12}-\frac{15960336537}{1321706623750}a^{11}-\frac{698549650887}{2643413247500}a^{10}+\frac{1055657228423}{2643413247500}a^{9}+\frac{263730105509}{660853311875}a^{8}-\frac{387800694743}{528682649500}a^{7}-\frac{6231438976827}{5286826495000}a^{6}+\frac{7211869938173}{5286826495000}a^{5}+\frac{1030858584297}{528682649500}a^{4}-\frac{1629936668981}{1057365299000}a^{3}-\frac{1694325175737}{2643413247500}a^{2}+\frac{66688918432}{132170662375}a+\frac{72857203833}{132170662375}$, $\frac{95116489387127}{74\!\cdots\!00}a^{15}-\frac{46443619251652}{928498903184375}a^{14}-\frac{971031233427991}{74\!\cdots\!00}a^{13}+\frac{59\!\cdots\!19}{37\!\cdots\!00}a^{12}-\frac{17\!\cdots\!81}{37\!\cdots\!00}a^{11}-\frac{930891652249221}{742799122547500}a^{10}+\frac{77\!\cdots\!59}{742799122547500}a^{9}+\frac{64\!\cdots\!67}{74\!\cdots\!00}a^{8}-\frac{48\!\cdots\!59}{18\!\cdots\!50}a^{7}-\frac{16\!\cdots\!43}{74\!\cdots\!00}a^{6}+\frac{23\!\cdots\!23}{74\!\cdots\!00}a^{5}+\frac{27\!\cdots\!61}{74\!\cdots\!00}a^{4}+\frac{78\!\cdots\!11}{18\!\cdots\!50}a^{3}-\frac{46\!\cdots\!43}{18\!\cdots\!50}a^{2}-\frac{32\!\cdots\!09}{371399561273750}a+\frac{851123221511654}{185699780636875}$, $\frac{1428128322163}{742799122547500}a^{15}-\frac{6266648540639}{742799122547500}a^{14}-\frac{23431486696733}{14\!\cdots\!00}a^{13}+\frac{371823213047259}{14\!\cdots\!00}a^{12}-\frac{154529929241537}{185699780636875}a^{11}+\frac{26359848848739}{148559824509500}a^{10}+\frac{123176429613801}{74279912254750}a^{9}+\frac{40937952091597}{185699780636875}a^{8}-\frac{31\!\cdots\!39}{742799122547500}a^{7}-\frac{15\!\cdots\!69}{14\!\cdots\!00}a^{6}+\frac{89\!\cdots\!29}{14\!\cdots\!00}a^{5}+\frac{392909910940811}{185699780636875}a^{4}-\frac{34\!\cdots\!33}{14\!\cdots\!00}a^{3}-\frac{539922782706777}{185699780636875}a^{2}+\frac{131633199518883}{74279912254750}a+\frac{43440071623212}{37139956127375}$, $\frac{4001019366753}{14\!\cdots\!00}a^{15}-\frac{17283028854147}{14\!\cdots\!00}a^{14}-\frac{16899085578449}{742799122547500}a^{13}+\frac{129231206325853}{371399561273750}a^{12}-\frac{3387224869281}{2971196490190}a^{11}+\frac{34907499558197}{185699780636875}a^{10}+\frac{410213278617492}{185699780636875}a^{9}+\frac{11\!\cdots\!99}{14\!\cdots\!00}a^{8}-\frac{93\!\cdots\!81}{14\!\cdots\!00}a^{7}-\frac{283800852783693}{148559824509500}a^{6}+\frac{13\!\cdots\!41}{14\!\cdots\!00}a^{5}+\frac{23\!\cdots\!13}{742799122547500}a^{4}-\frac{573565931194717}{185699780636875}a^{3}-\frac{34\!\cdots\!03}{742799122547500}a^{2}+\frac{20064339864859}{7427991225475}a-\frac{3088517021118}{37139956127375}$, $\frac{98004840774487}{74\!\cdots\!00}a^{15}-\frac{94072325294219}{18\!\cdots\!50}a^{14}-\frac{10\!\cdots\!61}{74\!\cdots\!00}a^{13}+\frac{30\!\cdots\!07}{18\!\cdots\!50}a^{12}-\frac{17\!\cdots\!21}{37\!\cdots\!00}a^{11}-\frac{264695311209543}{148559824509500}a^{10}+\frac{16\!\cdots\!17}{148559824509500}a^{9}+\frac{70\!\cdots\!87}{74\!\cdots\!00}a^{8}-\frac{25\!\cdots\!17}{928498903184375}a^{7}-\frac{18\!\cdots\!13}{74\!\cdots\!00}a^{6}+\frac{25\!\cdots\!53}{74\!\cdots\!00}a^{5}+\frac{30\!\cdots\!61}{74\!\cdots\!00}a^{4}+\frac{42\!\cdots\!97}{37\!\cdots\!00}a^{3}-\frac{24\!\cdots\!14}{928498903184375}a^{2}-\frac{28\!\cdots\!69}{371399561273750}a+\frac{870152183645509}{185699780636875}$, $\frac{3437068252916}{185699780636875}a^{15}-\frac{11546102791911}{148559824509500}a^{14}-\frac{243715710667827}{14\!\cdots\!00}a^{13}+\frac{35\!\cdots\!77}{14\!\cdots\!00}a^{12}-\frac{56\!\cdots\!41}{742799122547500}a^{11}+\frac{118926693939399}{185699780636875}a^{10}+\frac{26\!\cdots\!54}{185699780636875}a^{9}+\frac{122972646035881}{14855982450950}a^{8}-\frac{14\!\cdots\!97}{371399561273750}a^{7}-\frac{28\!\cdots\!83}{14\!\cdots\!00}a^{6}+\frac{14\!\cdots\!67}{297119649019000}a^{5}+\frac{25\!\cdots\!09}{742799122547500}a^{4}-\frac{83\!\cdots\!23}{14\!\cdots\!00}a^{3}-\frac{11\!\cdots\!21}{371399561273750}a^{2}+\frac{54856066833513}{37139956127375}a+\frac{97762234080568}{37139956127375}$, $\frac{108675684761917}{37\!\cdots\!00}a^{15}-\frac{433030352628691}{37\!\cdots\!00}a^{14}-\frac{21\!\cdots\!27}{74\!\cdots\!00}a^{13}+\frac{27\!\cdots\!71}{74\!\cdots\!00}a^{12}-\frac{10\!\cdots\!18}{928498903184375}a^{11}-\frac{280378472822361}{148559824509500}a^{10}+\frac{140781496592303}{5942392980380}a^{9}+\frac{65\!\cdots\!67}{37\!\cdots\!00}a^{8}-\frac{22\!\cdots\!51}{37\!\cdots\!00}a^{7}-\frac{34\!\cdots\!41}{74\!\cdots\!00}a^{6}+\frac{54\!\cdots\!71}{74\!\cdots\!00}a^{5}+\frac{70\!\cdots\!69}{928498903184375}a^{4}+\frac{39\!\cdots\!83}{74\!\cdots\!00}a^{3}-\frac{20\!\cdots\!67}{37\!\cdots\!00}a^{2}-\frac{26\!\cdots\!54}{185699780636875}a+\frac{18\!\cdots\!88}{185699780636875}$, $\frac{8395147667361}{928498903184375}a^{15}-\frac{153730086815617}{37\!\cdots\!00}a^{14}-\frac{61080252346893}{928498903184375}a^{13}+\frac{43\!\cdots\!11}{37\!\cdots\!00}a^{12}-\frac{76\!\cdots\!37}{18\!\cdots\!50}a^{11}+\frac{319384979083208}{185699780636875}a^{10}+\frac{12\!\cdots\!08}{185699780636875}a^{9}+\frac{35\!\cdots\!27}{18\!\cdots\!50}a^{8}-\frac{78\!\cdots\!27}{37\!\cdots\!00}a^{7}-\frac{36\!\cdots\!93}{18\!\cdots\!50}a^{6}+\frac{97\!\cdots\!51}{37\!\cdots\!00}a^{5}+\frac{33\!\cdots\!77}{37\!\cdots\!00}a^{4}-\frac{24\!\cdots\!67}{37\!\cdots\!00}a^{3}-\frac{13\!\cdots\!31}{928498903184375}a^{2}+\frac{964055567101432}{185699780636875}a-\frac{66444718934989}{185699780636875}$, $\frac{20689707178769}{37\!\cdots\!00}a^{15}-\frac{13724496847543}{928498903184375}a^{14}-\frac{584325199565249}{74\!\cdots\!00}a^{13}+\frac{45\!\cdots\!47}{74\!\cdots\!00}a^{12}-\frac{23\!\cdots\!97}{18\!\cdots\!50}a^{11}-\frac{461729628430289}{185699780636875}a^{10}+\frac{16\!\cdots\!79}{742799122547500}a^{9}+\frac{15\!\cdots\!57}{18\!\cdots\!50}a^{8}-\frac{13\!\cdots\!47}{37\!\cdots\!00}a^{7}-\frac{13\!\cdots\!07}{74\!\cdots\!00}a^{6}-\frac{33\!\cdots\!43}{74\!\cdots\!00}a^{5}+\frac{18\!\cdots\!43}{928498903184375}a^{4}+\frac{17\!\cdots\!01}{74\!\cdots\!00}a^{3}+\frac{12\!\cdots\!23}{18\!\cdots\!50}a^{2}-\frac{443418643564333}{185699780636875}a-\frac{218180556998294}{185699780636875}$, $\frac{94235158631543}{37\!\cdots\!00}a^{15}-\frac{86306672192486}{928498903184375}a^{14}-\frac{21\!\cdots\!63}{74\!\cdots\!00}a^{13}+\frac{23\!\cdots\!59}{74\!\cdots\!00}a^{12}-\frac{79\!\cdots\!52}{928498903184375}a^{11}-\frac{18\!\cdots\!59}{371399561273750}a^{10}+\frac{77\!\cdots\!81}{371399561273750}a^{9}+\frac{81\!\cdots\!03}{37\!\cdots\!00}a^{8}-\frac{18\!\cdots\!49}{37\!\cdots\!00}a^{7}-\frac{42\!\cdots\!99}{74\!\cdots\!00}a^{6}+\frac{42\!\cdots\!39}{74\!\cdots\!00}a^{5}+\frac{80\!\cdots\!06}{928498903184375}a^{4}+\frac{11\!\cdots\!17}{74\!\cdots\!00}a^{3}-\frac{17\!\cdots\!23}{37\!\cdots\!00}a^{2}-\frac{77\!\cdots\!37}{371399561273750}a+\frac{11\!\cdots\!97}{185699780636875}$, $\frac{198775114882663}{74\!\cdots\!00}a^{15}-\frac{336779190349837}{37\!\cdots\!00}a^{14}-\frac{11\!\cdots\!57}{37\!\cdots\!00}a^{13}+\frac{23\!\cdots\!97}{74\!\cdots\!00}a^{12}-\frac{30\!\cdots\!79}{37\!\cdots\!00}a^{11}-\frac{471983272419227}{74279912254750}a^{10}+\frac{99161340738127}{5942392980380}a^{9}+\frac{20\!\cdots\!13}{74\!\cdots\!00}a^{8}-\frac{14\!\cdots\!07}{37\!\cdots\!00}a^{7}-\frac{23\!\cdots\!81}{37\!\cdots\!00}a^{6}+\frac{10\!\cdots\!11}{37\!\cdots\!00}a^{5}+\frac{62\!\cdots\!39}{74\!\cdots\!00}a^{4}+\frac{36\!\cdots\!31}{74\!\cdots\!00}a^{3}-\frac{22\!\cdots\!47}{18\!\cdots\!50}a^{2}-\frac{22\!\cdots\!53}{185699780636875}a+\frac{433682285960541}{185699780636875}$
| 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: | \( 35504723.7489 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{4}\cdot 35504723.7489 \cdot 1}{2\cdot\sqrt{43149301773875176172265625}}\cr\approx \mathstrut & 1.07827291091 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80)
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^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80, 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^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80);
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^16 - 4*x^15 - 10*x^14 + 127*x^13 - 380*x^12 - 84*x^11 + 890*x^10 + 591*x^9 - 2244*x^8 - 1586*x^7 + 3018*x^6 + 2649*x^5 - 501*x^4 - 2078*x^3 - 204*x^2 + 520*x - 80);
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
$C_2^3:C_4$ (as 16T33):
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 32 |
| The 11 conjugacy class representatives for $C_2^3:C_4$ |
| Character table for $C_2^3:C_4$ |
Intermediate fields
| \(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), 4.4.51005.1 x2, 4.4.2525.1 x2, \(\Q(\sqrt{5}, \sqrt{101})\), 8.4.6568812813125.2 x2, 8.4.65037750625.1 x2, 8.8.65037750625.1 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(101\)
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.2.1.1 | $x^{2} + 101$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 101.4.3.2 | $x^{4} + 101$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 101.4.3.2 | $x^{4} + 101$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |