Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11)
gp: K = bnfinit(y^22 + 2*y^20 - 5*y^18 - 22*y^16 + 29*y^14 + 153*y^12 + 72*y^10 - 195*y^8 - 158*y^6 + 66*y^4 + 77*y^2 + 11, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11)
\( x^{22} + 2 x^{20} - 5 x^{18} - 22 x^{16} + 29 x^{14} + 153 x^{12} + 72 x^{10} - 195 x^{8} - 158 x^{6} + \cdots + 11 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-492568978448025181133471744\)
\(\medspace = -\,2^{18}\cdot 11^{5}\cdot 19^{4}\cdot 547^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.34\) | 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: |
\(2\), \(11\), \(19\), \(547\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{5528348828}a^{20}-\frac{272817717}{1382087207}a^{18}-\frac{1}{4}a^{17}-\frac{42819692}{1382087207}a^{16}-\frac{1}{4}a^{15}-\frac{123044705}{1382087207}a^{14}-\frac{250642240}{1382087207}a^{12}-\frac{1}{4}a^{11}-\frac{28171175}{1382087207}a^{10}-\frac{1}{4}a^{9}-\frac{392133315}{2764174414}a^{8}-\frac{439687842}{1382087207}a^{6}+\frac{1}{4}a^{5}-\frac{287650587}{5528348828}a^{4}-\frac{1}{4}a^{3}+\frac{359620784}{1382087207}a^{2}+\frac{1}{4}a+\frac{389481833}{5528348828}$, $\frac{1}{5528348828}a^{21}+\frac{290816339}{5528348828}a^{19}+\frac{1210808439}{5528348828}a^{17}-\frac{1}{4}a^{16}-\frac{123044705}{1382087207}a^{15}-\frac{1}{4}a^{14}+\frac{379518247}{5528348828}a^{13}+\frac{1269402507}{5528348828}a^{11}+\frac{1}{4}a^{10}+\frac{494976946}{1382087207}a^{9}+\frac{1}{4}a^{8}-\frac{376664161}{5528348828}a^{7}+\frac{273609155}{1382087207}a^{5}-\frac{1}{4}a^{4}-\frac{2707778485}{5528348828}a^{3}-\frac{1}{4}a^{2}+\frac{389481833}{5528348828}a+\frac{1}{4}$
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: | $10$ | 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: |
$\frac{792725095}{2764174414}a^{20}+\frac{845642949}{2764174414}a^{18}-\frac{4822819285}{2764174414}a^{16}-\frac{6523716598}{1382087207}a^{14}+\frac{35529305711}{2764174414}a^{12}+\frac{89448549059}{2764174414}a^{10}-\frac{14444533207}{1382087207}a^{8}-\frac{136607873999}{2764174414}a^{6}-\frac{122842295}{1382087207}a^{4}+\frac{62883390701}{2764174414}a^{2}+\frac{7405295673}{2764174414}$, $\frac{130212107}{1382087207}a^{20}+\frac{351159339}{2764174414}a^{18}-\frac{767186051}{1382087207}a^{16}-\frac{4778588097}{2764174414}a^{14}+\frac{10662317313}{2764174414}a^{12}+\frac{16619452600}{1382087207}a^{10}-\frac{2366231549}{2764174414}a^{8}-\frac{51700747649}{2764174414}a^{6}-\frac{5722998043}{1382087207}a^{4}+\frac{13661929029}{1382087207}a^{2}+\frac{8786648363}{2764174414}$, $\frac{82067133}{1382087207}a^{21}-\frac{90652948}{1382087207}a^{20}+\frac{226279087}{2764174414}a^{19}-\frac{131116289}{2764174414}a^{18}-\frac{1015796519}{2764174414}a^{17}+\frac{584425643}{1382087207}a^{16}-\frac{1507241424}{1382087207}a^{15}+\frac{1289707498}{1382087207}a^{14}+\frac{3482615631}{1382087207}a^{13}-\frac{4565247214}{1382087207}a^{12}+\frac{10765147189}{1382087207}a^{11}-\frac{17546434451}{2764174414}a^{10}-\frac{2089045921}{1382087207}a^{9}+\frac{6860215196}{1382087207}a^{8}-\frac{17113364174}{1382087207}a^{7}+\frac{13801524156}{1382087207}a^{6}-\frac{1351764775}{2764174414}a^{5}-\frac{10668085715}{2764174414}a^{4}+\frac{18187624031}{2764174414}a^{3}-\frac{13528845471}{2764174414}a^{2}+\frac{783669952}{1382087207}a+\frac{740922315}{2764174414}$, $\frac{621954672}{1382087207}a^{21}-\frac{82067133}{1382087207}a^{20}+\frac{1551404837}{2764174414}a^{19}-\frac{226279087}{2764174414}a^{18}-\frac{3677106611}{1382087207}a^{17}+\frac{1015796519}{2764174414}a^{16}-\frac{21863451503}{2764174414}a^{15}+\frac{1507241424}{1382087207}a^{14}+\frac{26093579181}{1382087207}a^{13}-\frac{3482615631}{1382087207}a^{12}+\frac{150788251185}{2764174414}a^{11}-\frac{10765147189}{1382087207}a^{10}-\frac{10524351043}{1382087207}a^{9}+\frac{2089045921}{1382087207}a^{8}-\frac{112389138020}{1382087207}a^{7}+\frac{17113364174}{1382087207}a^{6}-\frac{38202455867}{2764174414}a^{5}+\frac{1351764775}{2764174414}a^{4}+\frac{99849352315}{2764174414}a^{3}-\frac{18187624031}{2764174414}a^{2}+\frac{13555162035}{1382087207}a-\frac{783669952}{1382087207}$, $\frac{27809283}{5528348828}a^{21}-\frac{1519432397}{5528348828}a^{20}+\frac{13537877}{5528348828}a^{19}-\frac{449715975}{1382087207}a^{18}-\frac{35202689}{2764174414}a^{17}+\frac{8986223935}{5528348828}a^{16}-\frac{162222587}{5528348828}a^{15}+\frac{25904070557}{5528348828}a^{14}+\frac{1049568311}{5528348828}a^{13}-\frac{16233212638}{1382087207}a^{12}+\frac{122535089}{2764174414}a^{11}-\frac{177755223951}{5528348828}a^{10}+\frac{446478131}{5528348828}a^{9}+\frac{34174578911}{5528348828}a^{8}+\frac{9588286087}{5528348828}a^{7}+\frac{64135777840}{1382087207}a^{6}+\frac{12145524109}{5528348828}a^{5}+\frac{9722562645}{2764174414}a^{4}-\frac{2773659603}{2764174414}a^{3}-\frac{112801721585}{5528348828}a^{2}-\frac{3260679123}{2764174414}a-\frac{9891847765}{2764174414}$, $\frac{266134191}{5528348828}a^{21}+\frac{261542197}{5528348828}a^{20}+\frac{674345561}{5528348828}a^{19}+\frac{87840017}{1382087207}a^{18}-\frac{641237135}{2764174414}a^{17}-\frac{1484814403}{5528348828}a^{16}-\frac{6686484679}{5528348828}a^{15}-\frac{4716759661}{5528348828}a^{14}+\frac{6105464775}{5528348828}a^{13}+\frac{5183226471}{2764174414}a^{12}+\frac{23974784903}{2764174414}a^{11}+\frac{32290950357}{5528348828}a^{10}+\frac{28606710819}{5528348828}a^{9}-\frac{237743625}{5528348828}a^{8}-\frac{61771696103}{5528348828}a^{7}-\frac{11228544917}{1382087207}a^{6}-\frac{50415096753}{5528348828}a^{5}-\frac{5857307863}{2764174414}a^{4}+\frac{13686273363}{2764174414}a^{3}+\frac{17902842683}{5528348828}a^{2}+\frac{4760465919}{1382087207}a+\frac{1229656891}{2764174414}$, $\frac{1534733111}{5528348828}a^{21}+\frac{302451419}{5528348828}a^{20}+\frac{1943220303}{5528348828}a^{19}+\frac{67717701}{1382087207}a^{18}-\frac{2231851955}{1382087207}a^{17}-\frac{1784334025}{5528348828}a^{16}-\frac{26898164991}{5528348828}a^{15}-\frac{4562622843}{5528348828}a^{14}+\frac{63493844095}{5528348828}a^{13}+\frac{3462716722}{1382087207}a^{12}+\frac{46213422775}{1382087207}a^{11}+\frac{30194618897}{5528348828}a^{10}-\frac{20264675033}{5528348828}a^{9}-\frac{12479272523}{5528348828}a^{8}-\frac{262270403113}{5528348828}a^{7}-\frac{10098884160}{1382087207}a^{6}-\frac{37301012185}{5528348828}a^{5}+\frac{2266703663}{1382087207}a^{4}+\frac{57154517445}{2764174414}a^{3}+\frac{18835903649}{5528348828}a^{2}+\frac{6371911146}{1382087207}a-\frac{548029862}{1382087207}$, $\frac{505236041}{1382087207}a^{21}+\frac{597613211}{2764174414}a^{20}+\frac{2619299201}{5528348828}a^{19}+\frac{1400341497}{5528348828}a^{18}-\frac{11731800113}{5528348828}a^{17}-\frac{1735255708}{1382087207}a^{16}-\frac{8954908037}{1382087207}a^{15}-\frac{20124738905}{5528348828}a^{14}+\frac{82752009339}{5528348828}a^{13}+\frac{50487852617}{5528348828}a^{12}+\frac{246669753131}{5528348828}a^{11}+\frac{34201379778}{1382087207}a^{10}-\frac{10681018665}{2764174414}a^{9}-\frac{22539322209}{5528348828}a^{8}-\frac{349897878913}{5528348828}a^{7}-\frac{185635553667}{5528348828}a^{6}-\frac{61471116611}{5528348828}a^{5}-\frac{8034572391}{2764174414}a^{4}+\frac{148790390151}{5528348828}a^{3}+\frac{34224800965}{2764174414}a^{2}+\frac{8235314666}{1382087207}a+\frac{9672062269}{5528348828}$, $\frac{2385920}{19466017}a^{21}+\frac{1022360195}{5528348828}a^{20}+\frac{12239317}{77864068}a^{19}+\frac{1522753979}{5528348828}a^{18}-\frac{28620319}{38932034}a^{17}-\frac{2907160191}{2764174414}a^{16}-\frac{170462477}{77864068}a^{15}-\frac{19472737785}{5528348828}a^{14}+\frac{402421187}{77864068}a^{13}+\frac{39143793413}{5528348828}a^{12}+\frac{297235319}{19466017}a^{11}+\frac{33848084895}{1382087207}a^{10}-\frac{183503801}{77864068}a^{9}+\frac{8080078483}{5528348828}a^{8}-\frac{1845561793}{77864068}a^{7}-\frac{196756262851}{5528348828}a^{6}-\frac{68427910}{19466017}a^{5}-\frac{64917983493}{5528348828}a^{4}+\frac{485153443}{38932034}a^{3}+\frac{45318192341}{2764174414}a^{2}+\frac{249210469}{77864068}a+\frac{9064802172}{1382087207}$, $\frac{254714171}{2764174414}a^{20}+\frac{183315641}{2764174414}a^{18}-\frac{769284451}{1382087207}a^{16}-\frac{3556982363}{2764174414}a^{14}+\frac{12284360861}{2764174414}a^{12}+\frac{11600628190}{1382087207}a^{10}-\frac{14747669305}{2764174414}a^{8}-\frac{30590836639}{2764174414}a^{6}+\frac{13337950355}{2764174414}a^{4}+\frac{4866083003}{1382087207}a^{2}-\frac{2530486308}{1382087207}$
| 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: | \( 19198.3237336 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 19198.3237336 \cdot 1}{2\cdot\sqrt{492568978448025181133471744}}\cr\approx \mathstrut & 0.260602356534 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11)
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^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11, 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^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11);
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^22 + 2*x^20 - 5*x^18 - 22*x^16 + 29*x^14 + 153*x^12 + 72*x^10 - 195*x^8 - 158*x^6 + 66*x^4 + 77*x^2 + 11);
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
$C_2^{11}.A_{11}$ (as 22T52):
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 40874803200 |
| The 400 conjugacy class representatives for $C_2^{11}.A_{11}$ are not computed |
| Character table for $C_2^{11}.A_{11}$ is not computed |
Intermediate fields
| 11.3.836463893056.1 |
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 44 sibling: | 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 | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $22$ | R | ${\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | R | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.9.0.1}{9} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.18.18.119 | $x^{18} + 6 x^{14} - 4 x^{13} + 6 x^{12} + 16 x^{10} - 4 x^{9} + 28 x^{8} - 16 x^{7} + 20 x^{6} + 8 x^{5} + 32 x^{4} + 32 x^{2} - 16 x + 8$ | $6$ | $3$ | $18$ | 18T269 | $[4/3, 4/3, 4/3, 4/3, 4/3, 4/3]_{3}^{6}$ | |
|
\(11\)
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.5.1 | $x^{6} + 22$ | $6$ | $1$ | $5$ | $D_{6}$ | $[\ ]_{6}^{2}$ | |
| 11.6.0.1 | $x^{6} + 3 x^{4} + 4 x^{3} + 6 x^{2} + 7 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(19\)
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 19.4.2.1 | $x^{4} + 36 x^{3} + 366 x^{2} + 756 x + 6445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.14.0.1 | $x^{14} + 11 x^{7} + 11 x^{6} + 11 x^{5} + x^{4} + 5 x^{3} + 16 x^{2} + 7 x + 2$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
|
\(547\)
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{547}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ |