Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1)
gp: K = bnfinit(y^16 - 2*y^15 - 19*y^14 + 26*y^13 + 130*y^12 - 67*y^11 - 463*y^10 - 207*y^9 + 1033*y^8 + 924*y^7 - 1394*y^6 - 803*y^5 + 937*y^4 + 5*y^3 - 100*y^2 + 19*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1)
\( x^{16} - 2 x^{15} - 19 x^{14} + 26 x^{13} + 130 x^{12} - 67 x^{11} - 463 x^{10} - 207 x^{9} + 1033 x^{8} + \cdots - 1 \)
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: | $[14, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-170690133797345543359375\)
\(\medspace = -\,5^{8}\cdot 19^{4}\cdot 31^{3}\cdot 103^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(28.31\) | 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: | $5^{1/2}19^{1/2}31^{1/2}103^{1/2}\approx 550.7585677953634$ | ||
| Ramified primes: |
\(5\), \(19\), \(31\), \(103\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3}a^{14}-\frac{1}{3}a^{11}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{4}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{18954698577}a^{15}+\frac{1475697238}{18954698577}a^{14}+\frac{2470611381}{6318232859}a^{13}-\frac{2331820093}{18954698577}a^{12}-\frac{6277501706}{18954698577}a^{11}-\frac{9181202648}{18954698577}a^{10}-\frac{4260112628}{18954698577}a^{9}-\frac{9293485615}{18954698577}a^{8}-\frac{1905171451}{6318232859}a^{7}+\frac{2061685955}{6318232859}a^{6}+\frac{2020343407}{18954698577}a^{5}+\frac{638166868}{18954698577}a^{4}+\frac{5382831338}{18954698577}a^{3}-\frac{1468390369}{18954698577}a^{2}-\frac{2465593538}{18954698577}a+\frac{569967328}{18954698577}$
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: | $14$ | 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: |
$\frac{550803195}{6188279}a^{15}-\frac{1003523385}{6188279}a^{14}-\frac{10645966073}{6188279}a^{13}+\frac{12427001971}{6188279}a^{12}+\frac{73857749964}{6188279}a^{11}-\frac{23761219860}{6188279}a^{10}-\frac{259530512950}{6188279}a^{9}-\frac{160390005139}{6188279}a^{8}+\frac{541225271829}{6188279}a^{7}+\frac{606615459500}{6188279}a^{6}-\frac{660633690622}{6188279}a^{5}-\frac{562892254635}{6188279}a^{4}+\frac{415679752862}{6188279}a^{3}+\frac{78695706691}{6188279}a^{2}-\frac{40886326775}{6188279}a+\frac{3028743148}{6188279}$, $\frac{278002412710}{6318232859}a^{15}-\frac{507237853303}{6318232859}a^{14}-\frac{5373198741943}{6318232859}a^{13}+\frac{6289003271821}{6318232859}a^{12}+\frac{37284073216517}{6318232859}a^{11}-\frac{12121204159961}{6318232859}a^{10}-\frac{131102890363805}{6318232859}a^{9}-\frac{80568341215552}{6318232859}a^{8}+\frac{273841783471405}{6318232859}a^{7}+\frac{305845470255059}{6318232859}a^{6}-\frac{335105877323051}{6318232859}a^{5}-\frac{284232126559503}{6318232859}a^{4}+\frac{211210448946066}{6318232859}a^{3}+\frac{39636253739569}{6318232859}a^{2}-\frac{20828878034031}{6318232859}a+\frac{1556179711143}{6318232859}$, $\frac{647003513560}{6318232859}a^{15}-\frac{1176261325562}{6318232859}a^{14}-\frac{12506909043511}{6318232859}a^{13}+\frac{14544472806326}{6318232859}a^{12}+\frac{86755920215744}{6318232859}a^{11}-\frac{27537220906173}{6318232859}a^{10}-\frac{304580079300203}{6318232859}a^{9}-\frac{189474283636235}{6318232859}a^{8}+\frac{633858619989329}{6318232859}a^{7}+\frac{713467107736025}{6318232859}a^{6}-\frac{771717562158353}{6318232859}a^{5}-\frac{660571798263952}{6318232859}a^{4}+\frac{485363494826618}{6318232859}a^{3}+\frac{92173531364291}{6318232859}a^{2}-\frac{47812403351500}{6318232859}a+\frac{3527528260078}{6318232859}$, $\frac{724044699883}{6318232859}a^{15}-\frac{1309056781248}{6318232859}a^{14}-\frac{14005145904607}{6318232859}a^{13}+\frac{16133501755233}{6318232859}a^{12}+\frac{97160522790151}{6318232859}a^{11}-\frac{29848017668175}{6318232859}a^{10}-\frac{340521852747247}{6318232859}a^{9}-\frac{214972311921440}{6318232859}a^{8}+\frac{705378442678843}{6318232859}a^{7}+\frac{802308151286474}{6318232859}a^{6}-\frac{854054677877517}{6318232859}a^{5}-\frac{740473139497628}{6318232859}a^{4}+\frac{536762358418977}{6318232859}a^{3}+\frac{103382172887463}{6318232859}a^{2}-\frac{52842279773943}{6318232859}a+\frac{3868857722579}{6318232859}$, $\frac{794071606396}{6318232859}a^{15}-\frac{1445078119864}{6318232859}a^{14}-\frac{15349278993174}{6318232859}a^{13}+\frac{17880721406506}{6318232859}a^{12}+\frac{106485242114426}{6318232859}a^{11}-\frac{34005135556090}{6318232859}a^{10}-\frac{374034303893310}{6318232859}a^{9}-\frac{232003831860941}{6318232859}a^{8}+\frac{779189125826516}{6318232859}a^{7}+\frac{875514807159110}{6318232859}a^{6}-\frac{949626576834080}{6318232859}a^{5}-\frac{811918444928865}{6318232859}a^{4}+\frac{597000820353783}{6318232859}a^{3}+\frac{113817731026129}{6318232859}a^{2}-\frac{58644287364191}{6318232859}a+\frac{4330416538828}{6318232859}$, $\frac{2761583964148}{18954698577}a^{15}-\frac{5010675904352}{18954698577}a^{14}-\frac{17799219227978}{6318232859}a^{13}+\frac{61888783942301}{18954698577}a^{12}+\frac{370446448571362}{18954698577}a^{11}-\frac{116255678418686}{18954698577}a^{10}-\frac{12\!\cdots\!91}{18954698577}a^{9}-\frac{812769806648752}{18954698577}a^{8}+\frac{900338290069589}{6318232859}a^{7}+\frac{10\!\cdots\!95}{6318232859}a^{6}-\frac{32\!\cdots\!96}{18954698577}a^{5}-\frac{28\!\cdots\!13}{18954698577}a^{4}+\frac{20\!\cdots\!80}{18954698577}a^{3}+\frac{395086633716836}{18954698577}a^{2}-\frac{203017613680676}{18954698577}a+\frac{14907330148255}{18954698577}$, $\frac{1241326470496}{18954698577}a^{15}-\frac{2257663681607}{18954698577}a^{14}-\frac{7998147609107}{6318232859}a^{13}+\frac{27922261487186}{18954698577}a^{12}+\frac{166442316654763}{18954698577}a^{11}-\frac{52945683937754}{18954698577}a^{10}-\frac{584438615296913}{18954698577}a^{9}-\frac{363220246958926}{18954698577}a^{8}+\frac{405571291280552}{6318232859}a^{7}+\frac{456220913555431}{6318232859}a^{6}-\frac{14\!\cdots\!04}{18954698577}a^{5}-\frac{12\!\cdots\!18}{18954698577}a^{4}+\frac{931580327471111}{18954698577}a^{3}+\frac{177360279699311}{18954698577}a^{2}-\frac{91620753611498}{18954698577}a+\frac{6748269502180}{18954698577}$, $\frac{4532907375625}{18954698577}a^{15}-\frac{8240758079782}{18954698577}a^{14}-\frac{29210775489497}{6318232859}a^{13}+\frac{101915784591899}{18954698577}a^{12}+\frac{202654575550147}{6318232859}a^{11}-\frac{64382318424425}{6318232859}a^{10}-\frac{711593310060412}{6318232859}a^{9}-\frac{13\!\cdots\!81}{18954698577}a^{8}+\frac{14\!\cdots\!94}{6318232859}a^{7}+\frac{16\!\cdots\!18}{6318232859}a^{6}-\frac{54\!\cdots\!81}{18954698577}a^{5}-\frac{46\!\cdots\!59}{18954698577}a^{4}+\frac{34\!\cdots\!30}{18954698577}a^{3}+\frac{647187077480632}{18954698577}a^{2}-\frac{335371767268763}{18954698577}a+\frac{24723096929687}{18954698577}$, $\frac{377875993061}{18954698577}a^{15}-\frac{675986078245}{18954698577}a^{14}-\frac{2438926658567}{6318232859}a^{13}+\frac{8275445371615}{18954698577}a^{12}+\frac{50757940283453}{18954698577}a^{11}-\frac{14583130318486}{18954698577}a^{10}-\frac{177257324561638}{18954698577}a^{9}-\frac{115164403248740}{18954698577}a^{8}+\frac{121261979233156}{6318232859}a^{7}+\frac{140753234400302}{6318232859}a^{6}-\frac{435385695401113}{18954698577}a^{5}-\frac{386947555609042}{18954698577}a^{4}+\frac{273054866358238}{18954698577}a^{3}+\frac{54034347573289}{18954698577}a^{2}-\frac{26856859598713}{18954698577}a+\frac{1925415439133}{18954698577}$, $a$, $\frac{174711607820}{18954698577}a^{15}-\frac{98261669293}{6318232859}a^{14}-\frac{1133911784752}{6318232859}a^{13}+\frac{3471615531985}{18954698577}a^{12}+\frac{23591665510831}{18954698577}a^{11}-\frac{4319860616540}{18954698577}a^{10}-\frac{80795447057348}{18954698577}a^{9}-\frac{20135986228645}{6318232859}a^{8}+\frac{52477966403907}{6318232859}a^{7}+\frac{67813746585017}{6318232859}a^{6}-\frac{176280207592153}{18954698577}a^{5}-\frac{59693903074650}{6318232859}a^{4}+\frac{109959326216011}{18954698577}a^{3}+\frac{8114475419627}{6318232859}a^{2}-\frac{11042266438339}{18954698577}a+\frac{248560588911}{6318232859}$, $\frac{1585395211187}{18954698577}a^{15}-\frac{964886753039}{6318232859}a^{14}-\frac{10208002565315}{6318232859}a^{13}+\frac{35860431566440}{18954698577}a^{12}+\frac{212348000526484}{18954698577}a^{11}-\frac{68824420722932}{18954698577}a^{10}-\frac{746321003968244}{18954698577}a^{9}-\frac{153433631219800}{6318232859}a^{8}+\frac{519289681114074}{6318232859}a^{7}+\frac{580703727750949}{6318232859}a^{6}-\frac{19\!\cdots\!71}{18954698577}a^{5}-\frac{538891304964377}{6318232859}a^{4}+\frac{11\!\cdots\!57}{18954698577}a^{3}+\frac{75947190601553}{6318232859}a^{2}-\frac{116760281832826}{18954698577}a+\frac{2854307122602}{6318232859}$, $\frac{4589615746435}{18954698577}a^{15}-\frac{2772259210862}{6318232859}a^{14}-\frac{29583926504337}{6318232859}a^{13}+\frac{102639723837179}{18954698577}a^{12}+\frac{615658763560313}{18954698577}a^{11}-\frac{191752520078866}{18954698577}a^{10}-\frac{21\!\cdots\!83}{18954698577}a^{9}-\frac{451555833279768}{6318232859}a^{8}+\frac{14\!\cdots\!90}{6318232859}a^{7}+\frac{16\!\cdots\!49}{6318232859}a^{6}-\frac{54\!\cdots\!18}{18954698577}a^{5}-\frac{15\!\cdots\!33}{6318232859}a^{4}+\frac{34\!\cdots\!32}{18954698577}a^{3}+\frac{217412380184716}{6318232859}a^{2}-\frac{337357391271767}{18954698577}a+\frac{8298254566366}{6318232859}$, $\frac{445783715789}{18954698577}a^{15}-\frac{816128446648}{18954698577}a^{14}-\frac{2871383751426}{6318232859}a^{13}+\frac{10141645549987}{18954698577}a^{12}+\frac{59783971484969}{18954698577}a^{11}-\frac{19834932493426}{18954698577}a^{10}-\frac{210503345868937}{18954698577}a^{9}-\frac{128054338781831}{18954698577}a^{8}+\frac{147019711256857}{6318232859}a^{7}+\frac{163133470594069}{6318232859}a^{6}-\frac{541769194983082}{18954698577}a^{5}-\frac{456223274355187}{18954698577}a^{4}+\frac{341646755545195}{18954698577}a^{3}+\frac{63684669895222}{18954698577}a^{2}-\frac{33595110601750}{18954698577}a+\frac{2528770760144}{18954698577}$
| 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: | \( 1497538.20772 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{1}\cdot 1497538.20772 \cdot 1}{2\cdot\sqrt{170690133797345543359375}}\cr\approx \mathstrut & 0.186570731470 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1)
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 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1, 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 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1);
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 - 2*x^15 - 19*x^14 + 26*x^13 + 130*x^12 - 67*x^11 - 463*x^10 - 207*x^9 + 1033*x^8 + 924*x^7 - 1394*x^6 - 803*x^5 + 937*x^4 + 5*x^3 - 100*x^2 + 19*x - 1);
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
$C_{2440}.D_6$ (as 16T1759):
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 12288 |
| The 93 conjugacy class representatives for $C_{2440}.D_6$ |
| Character table for $C_{2440}.D_6$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.1957.1, 8.8.2393655625.1 |
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
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.4.8809813357282350625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.8.0.1}{8} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{6}$ | R | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.16.8.1 | $x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
|
\(19\)
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 31.6.0.1 | $x^{6} + 19 x^{3} + 16 x^{2} + 8 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 31.6.3.1 | $x^{6} + 961 x^{2} - 834148$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(103\)
| 103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 103.4.0.1 | $x^{4} + 2 x^{2} + 88 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 103.8.4.1 | $x^{8} + 416 x^{6} + 176 x^{5} + 64080 x^{4} - 35904 x^{3} + 4330880 x^{2} - 5564416 x + 109124096$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |