Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832)
gp: K = bnfinit(y^21 - 63*y^15 - 54*y^14 + 9261*y^3 + 23814*y^2 + 20412*y + 5832, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832)
\( x^{21} - 63x^{15} - 54x^{14} + 9261x^{3} + 23814x^{2} + 20412x + 5832 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $21$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[7, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-42806290740019959803859400295965724054212608\)
\(\medspace = -\,2^{12}\cdot 3^{44}\cdot 7^{21}\cdot 19\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(119.59\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-133}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{3}a^{8}$, $\frac{1}{3}a^{9}$, $\frac{1}{9}a^{10}+\frac{1}{3}a^{4}$, $\frac{1}{9}a^{11}+\frac{1}{3}a^{5}$, $\frac{1}{9}a^{12}$, $\frac{1}{27}a^{13}-\frac{1}{9}a^{9}-\frac{1}{9}a^{7}+\frac{1}{3}a^{5}-\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{27}a^{14}-\frac{1}{9}a^{8}+\frac{1}{3}a^{2}$, $\frac{1}{27}a^{15}-\frac{1}{9}a^{9}+\frac{1}{3}a^{3}$, $\frac{1}{54}a^{16}-\frac{1}{54}a^{14}-\frac{1}{54}a^{13}-\frac{1}{18}a^{12}-\frac{1}{18}a^{10}+\frac{1}{18}a^{9}+\frac{1}{18}a^{8}-\frac{1}{9}a^{7}-\frac{1}{6}a^{5}-\frac{1}{3}a^{4}+\frac{1}{6}a^{3}-\frac{1}{6}a^{2}-\frac{1}{6}a$, $\frac{1}{54}a^{17}-\frac{1}{54}a^{15}-\frac{1}{54}a^{14}-\frac{1}{54}a^{13}-\frac{1}{18}a^{11}-\frac{1}{18}a^{10}-\frac{1}{18}a^{9}-\frac{1}{9}a^{8}-\frac{1}{9}a^{7}-\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}a$, $\frac{1}{54}a^{18}-\frac{1}{54}a^{15}-\frac{1}{54}a^{13}-\frac{1}{18}a^{11}-\frac{1}{18}a^{9}-\frac{1}{6}a^{8}+\frac{1}{18}a^{7}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{2519424}a^{19}-\frac{1111}{209952}a^{18}+\frac{97}{23328}a^{17}+\frac{17}{2916}a^{16}-\frac{7}{1944}a^{15}-\frac{7}{279936}a^{13}+\frac{1}{46656}a^{12}+\frac{185}{7776}a^{11}+\frac{25}{1296}a^{10}-\frac{19}{216}a^{9}+\frac{5}{36}a^{8}+\frac{1}{18}a^{7}-\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}+\frac{1}{6}a^{2}+\frac{31447}{93312}a+\frac{49}{15552}$, $\frac{1}{235092492288}a^{20}+\frac{6665}{39182082048}a^{19}+\frac{44422225}{6530347008}a^{18}-\frac{8578855}{1088391168}a^{17}-\frac{580703}{181398528}a^{16}-\frac{550231}{30233088}a^{15}-\frac{179900359}{26121388032}a^{14}+\frac{587}{52488}a^{13}-\frac{431}{8748}a^{12}-\frac{31}{1458}a^{11}+\frac{5}{486}a^{10}+\frac{23}{162}a^{9}-\frac{7}{54}a^{8}-\frac{1}{6}a^{5}+\frac{1}{6}a^{3}+\frac{1451188567}{8707129344}a^{2}-\frac{119789231}{725594112}a+\frac{326599}{241864704}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
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: | $13$ | 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: |
$\frac{84081269671}{235092492288}a^{20}-\frac{12272992417}{39182082048}a^{19}+\frac{1790624983}{6530347008}a^{18}-\frac{261137905}{1088391168}a^{17}+\frac{38067463}{181398528}a^{16}-\frac{5547073}{30233088}a^{15}-\frac{584380262353}{26121388032}a^{14}+\frac{117649}{419904}a^{13}-\frac{16807}{69984}a^{12}+\frac{2401}{11664}a^{11}-\frac{343}{1944}a^{10}+\frac{49}{324}a^{9}-\frac{7}{54}a^{8}+\frac{1}{9}a^{7}+\frac{28839875497153}{8707129344}a^{2}+\frac{4075155121303}{725594112}a+\frac{576003011665}{241864704}$, $\frac{823543}{15116544}a^{20}-\frac{16807}{419904}a^{19}+\frac{12005}{419904}a^{18}-\frac{343}{17496}a^{17}+\frac{49}{3888}a^{16}-\frac{7}{972}a^{15}-\frac{5759617}{1679616}a^{14}-\frac{117649}{279936}a^{13}+\frac{16807}{46656}a^{12}-\frac{2401}{7776}a^{11}+\frac{343}{1296}a^{10}-\frac{49}{216}a^{9}+\frac{7}{36}a^{8}-\frac{1}{6}a^{7}+\frac{282475249}{559872}a^{2}+\frac{28824005}{31104}a+\frac{823543}{1944}$, $\frac{126143405969}{29386561536}a^{20}-\frac{18306958295}{4897760256}a^{19}+\frac{2595738113}{816293376}a^{18}-\frac{362732135}{136048896}a^{17}+\frac{51966065}{22674816}a^{16}-\frac{8217431}{3779136}a^{15}-\frac{875196400823}{3265173504}a^{14}+\frac{332329}{419904}a^{13}+\frac{344657}{69984}a^{12}-\frac{83951}{11664}a^{11}+\frac{3893}{1944}a^{10}+\frac{4327}{324}a^{9}-\frac{2011}{54}a^{8}+\frac{548}{9}a^{7}-\frac{407}{6}a^{6}+38a^{5}+\frac{253}{6}a^{4}-\frac{989}{6}a^{3}+\frac{43578649519943}{1088391168}a^{2}+\frac{6102661595033}{90699264}a+\frac{853457152967}{30233088}$, $\frac{13405295405}{117546246144}a^{20}-\frac{2425806539}{19591041024}a^{19}+\frac{468870269}{3265173504}a^{18}-\frac{89804027}{544195584}a^{17}+\frac{18728909}{90699264}a^{16}-\frac{3689771}{15116544}a^{15}-\frac{89647581947}{13060694016}a^{14}+\frac{1055165}{839808}a^{13}-\frac{258491}{139968}a^{12}+\frac{46925}{23328}a^{11}-\frac{12659}{3888}a^{10}+\frac{2081}{648}a^{9}-\frac{605}{108}a^{8}+\frac{97}{18}a^{7}-9a^{6}+\frac{19}{2}a^{5}-15a^{4}+\frac{83}{6}a^{3}+\frac{4466683789643}{4353564672}a^{2}+\frac{573314624621}{362797056}a+\frac{76428353435}{120932352}$, $\frac{1353127939}{6530347008}a^{20}-\frac{202385125}{1088391168}a^{19}+\frac{29798131}{181398528}a^{18}-\frac{4394869}{30233088}a^{17}+\frac{666595}{5038848}a^{16}-\frac{103621}{839808}a^{15}-\frac{9399635797}{725594112}a^{14}+\frac{7657}{17496}a^{13}-\frac{631}{2916}a^{12}+\frac{47}{243}a^{11}-\frac{77}{162}a^{10}+\frac{31}{54}a^{9}+\frac{5}{9}a^{8}-\frac{3}{2}a^{6}+\frac{4}{3}a^{5}+\frac{29}{6}a^{4}+\frac{3}{2}a^{3}+\frac{462308897797}{241864704}a^{2}+\frac{64806320755}{20155392}a+\frac{9087386197}{6718464}$, $\frac{641494037603}{235092492288}a^{20}-\frac{84629708549}{39182082048}a^{19}+\frac{11112881747}{6530347008}a^{18}-\frac{1449206165}{1088391168}a^{17}+\frac{187333955}{181398528}a^{16}-\frac{23982629}{30233088}a^{15}-\frac{4474597778933}{26121388032}a^{14}-\frac{4927369}{419904}a^{13}+\frac{683359}{69984}a^{12}-\frac{96049}{11664}a^{11}+\frac{13567}{1944}a^{10}-\frac{1897}{324}a^{9}+\frac{125}{27}a^{8}-\frac{35}{9}a^{7}+\frac{7}{2}a^{6}-\frac{23}{6}a^{5}+\frac{19}{6}a^{4}-\frac{10}{3}a^{3}+\frac{220029552521381}{8707129344}a^{2}+\frac{32636421409427}{725594112}a+\frac{4843009801589}{241864704}$, $\frac{78460291969}{19591041024}a^{20}-\frac{11276748487}{3265173504}a^{19}+\frac{1620080689}{544195584}a^{18}-\frac{232639639}{90699264}a^{17}+\frac{33428449}{15116544}a^{16}-\frac{4810855}{2519424}a^{15}-\frac{545625669703}{2176782336}a^{14}-\frac{36557}{279936}a^{13}+\frac{11147}{46656}a^{12}-\frac{2333}{7776}a^{11}+\frac{35}{1296}a^{10}+\frac{91}{216}a^{9}-\frac{19}{36}a^{8}+\frac{2}{3}a^{7}-\frac{3}{2}a^{6}+a^{5}+3a^{4}-\frac{35}{6}a^{3}+\frac{26913331333591}{725594112}a^{2}+\frac{3832838861113}{60466176}a+\frac{545822945527}{20155392}$, $\frac{58549808172847}{235092492288}a^{20}-\frac{8421861780889}{39182082048}a^{19}+\frac{1211401538911}{6530347008}a^{18}-\frac{174247810537}{1088391168}a^{17}+\frac{25063705231}{181398528}a^{16}-\frac{3605083705}{30233088}a^{15}-\frac{407160660815497}{26121388032}a^{14}+\frac{3213047}{839808}a^{13}-\frac{456785}{139968}a^{12}+\frac{66551}{23328}a^{11}-\frac{9785}{3888}a^{10}+\frac{1403}{648}a^{9}-\frac{197}{108}a^{8}+\frac{3}{2}a^{7}-\frac{7}{6}a^{6}+\frac{2}{3}a^{5}-\frac{7}{6}a^{4}+\frac{13}{6}a^{3}+\frac{20\!\cdots\!97}{8707129344}a^{2}+\frac{28\!\cdots\!23}{725594112}a+\frac{407044227009097}{241864704}$, $\frac{17313390807593}{78364164096}a^{20}-\frac{2489205622223}{13060694016}a^{19}+\frac{357880262777}{2176782336}a^{18}-\frac{51453719807}{362797056}a^{17}+\frac{7397612489}{60466176}a^{16}-\frac{1063394351}{10077696}a^{15}-\frac{120401686015199}{8707129344}a^{14}-\frac{517189}{279936}a^{13}+\frac{72403}{46656}a^{12}-\frac{10405}{7776}a^{11}+\frac{1867}{1296}a^{10}-\frac{553}{216}a^{9}+\frac{179}{36}a^{8}-\frac{43}{6}a^{7}+\frac{35}{6}a^{6}+\frac{5}{3}a^{5}-\frac{25}{2}a^{4}+\frac{95}{6}a^{3}+\frac{59\!\cdots\!71}{2902376448}a^{2}+\frac{845634492688025}{241864704}a+\frac{120420991246751}{80621568}$, $\frac{1650586603259}{117546246144}a^{20}-\frac{241437520205}{19591041024}a^{19}+\frac{35321387819}{3265173504}a^{18}-\frac{5168847773}{544195584}a^{17}+\frac{756825947}{90699264}a^{16}-\frac{110640749}{15116544}a^{15}-\frac{11470182099485}{13060694016}a^{14}+\frac{10451207}{839808}a^{13}-\frac{1551905}{139968}a^{12}+\frac{231143}{23328}a^{11}-\frac{33761}{3888}a^{10}+\frac{4895}{648}a^{9}-\frac{743}{108}a^{8}+\frac{119}{18}a^{7}-\frac{31}{6}a^{6}+\frac{9}{2}a^{5}-\frac{31}{6}a^{4}+\frac{8}{3}a^{3}+\frac{566146851353165}{4353564672}a^{2}+\frac{79915389993467}{362797056}a+\frac{11286964814333}{120932352}$, $\frac{42014450377}{235092492288}a^{20}-\frac{6250620847}{39182082048}a^{19}+\frac{864285721}{6530347008}a^{18}-\frac{130481119}{1088391168}a^{17}+\frac{21492841}{181398528}a^{16}-\frac{2003983}{30233088}a^{15}-\frac{292202434879}{26121388032}a^{14}+\frac{247757}{839808}a^{13}+\frac{27925}{139968}a^{12}+\frac{8045}{23328}a^{11}-\frac{2939}{3888}a^{10}-\frac{1195}{648}a^{9}-\frac{65}{108}a^{8}+\frac{11}{6}a^{7}+\frac{17}{2}a^{6}+\frac{37}{6}a^{5}-\frac{55}{6}a^{4}-\frac{85}{3}a^{3}+\frac{14091695070031}{8707129344}a^{2}+\frac{2036276330233}{725594112}a+\frac{291112742527}{241864704}$, $\frac{195249193673}{19591041024}a^{20}-\frac{27090218207}{3265173504}a^{19}+\frac{3759292601}{544195584}a^{18}-\frac{521564783}{90699264}a^{17}+\frac{72372521}{15116544}a^{16}-\frac{10051775}{2519424}a^{15}-\frac{1359487455551}{2176782336}a^{14}-\frac{5120471}{279936}a^{13}+\frac{709649}{46656}a^{12}-\frac{99095}{7776}a^{11}+\frac{13817}{1296}a^{10}-\frac{1943}{216}a^{9}+\frac{83}{12}a^{8}-\frac{50}{9}a^{7}+5a^{6}-\frac{9}{2}a^{5}+\frac{35}{6}a^{4}-\frac{5}{6}a^{3}+\frac{66969505971023}{725594112}a^{2}+\frac{9705185954129}{60466176}a+\frac{1407438585935}{20155392}$, $\frac{48690247}{3673320192}a^{20}-\frac{1843099}{612220032}a^{19}-\frac{1739645}{102036672}a^{18}+\frac{239009}{17006112}a^{17}+\frac{50767}{2834352}a^{16}-\frac{36343}{472392}a^{15}-\frac{327234097}{408146688}a^{14}-\frac{169589}{419904}a^{13}+\frac{70883}{69984}a^{12}+\frac{1075}{11664}a^{11}-\frac{2437}{1944}a^{10}+\frac{1045}{324}a^{9}+\frac{49}{27}a^{8}-\frac{131}{18}a^{7}+\frac{16}{3}a^{6}+5a^{5}-\frac{217}{6}a^{4}+2a^{3}+\frac{21031644577}{136048896}a^{2}+\frac{532697611}{2834352}a+\frac{251229367}{3779136}$
| 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: | \( 227380547912000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 227380547912000 \cdot 1}{2\cdot\sqrt{42806290740019959803859400295965724054212608}}\cr\approx \mathstrut & 0.859881415500569 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832)
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^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832, 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^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832);
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^21 - 63*x^15 - 54*x^14 + 9261*x^3 + 23814*x^2 + 20412*x + 5832);
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
$S_7\wr C_3$ (as 21T159):
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 non-solvable group of order 384072192000 |
| The 1165 conjugacy class representatives for $S_7\wr C_3$ |
| Character table for $S_7\wr C_3$ |
Intermediate fields
| \(\Q(\zeta_{9})^+\) |
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
| Degree 42 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | ${\href{/padicField/13.9.0.1}{9} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | R | $18{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.9.0.1}{9} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.3.0.1}{3} }^{3}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.9.0.1}{9} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 2.12.12.8 | $x^{12} - 4 x^{11} + 20 x^{10} - 64 x^{9} + 196 x^{8} - 336 x^{7} + 736 x^{6} + 160 x^{5} + 1712 x^{4} + 5632 x^{3} + 5440 x^{2} + 6784 x + 5824$ | $2$ | $6$ | $12$ | 12T51 | $[2, 2, 2, 2]^{6}$ | |
|
\(3\)
| 3.3.4.2 | $x^{3} + 6 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ |
| Deg $18$ | $18$ | $1$ | $40$ | ||||
|
\(7\)
| Deg $21$ | $7$ | $3$ | $21$ | |||
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.6.0.1 | $x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |