Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352)
gp: K = bnfinit(y^20 - 8*y^19 + 34*y^18 - 116*y^17 + 338*y^16 - 836*y^15 + 1840*y^14 - 3640*y^13 + 6408*y^12 - 10176*y^11 + 14560*y^10 - 18560*y^9 + 21128*y^8 - 21328*y^7 + 18704*y^6 - 14048*y^5 + 8976*y^4 - 4576*y^3 + 1760*y^2 - 704*y + 352, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352)
\( x^{20} - 8 x^{19} + 34 x^{18} - 116 x^{17} + 338 x^{16} - 836 x^{15} + 1840 x^{14} - 3640 x^{13} + \cdots + 352 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(49338146756019243307761664\)
\(\medspace = 2^{30}\cdot 11^{16}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(19.26\) | 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: | $2^{15/8}11^{4/5}\approx 24.977294240287762$ | ||
| Ramified primes: |
\(2\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{688}a^{18}+\frac{19}{688}a^{17}+\frac{7}{344}a^{16}-\frac{9}{344}a^{15}-\frac{21}{344}a^{14}-\frac{15}{344}a^{13}+\frac{3}{86}a^{12}+\frac{3}{172}a^{11}-\frac{5}{86}a^{10}+\frac{4}{43}a^{9}-\frac{15}{172}a^{8}-\frac{7}{43}a^{7}+\frac{2}{43}a^{6}+\frac{1}{43}a^{5}+\frac{1}{43}a^{4}-\frac{8}{43}a^{3}-\frac{16}{43}a^{2}+\frac{20}{43}a+\frac{19}{43}$, $\frac{1}{3406356112}a^{19}-\frac{625525}{1703178056}a^{18}+\frac{29140343}{3406356112}a^{17}+\frac{39878045}{1703178056}a^{16}+\frac{5102225}{212897257}a^{15}-\frac{23406541}{1703178056}a^{14}-\frac{1161701}{1703178056}a^{13}+\frac{5472279}{851589028}a^{12}+\frac{36719243}{851589028}a^{11}-\frac{73021841}{851589028}a^{10}-\frac{15395449}{425794514}a^{9}-\frac{30789639}{425794514}a^{8}+\frac{6041046}{212897257}a^{7}-\frac{46713705}{212897257}a^{6}+\frac{64208451}{425794514}a^{5}-\frac{2045967}{425794514}a^{4}-\frac{9305043}{212897257}a^{3}-\frac{14462082}{212897257}a^{2}+\frac{14495725}{212897257}a-\frac{53617858}{212897257}$
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 $1$
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: | $9$ | 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: |
$\frac{23282539}{1703178056}a^{19}-\frac{171319277}{1703178056}a^{18}+\frac{629459989}{1703178056}a^{17}-\frac{3737834429}{3406356112}a^{16}+\frac{2466299073}{851589028}a^{15}-\frac{10847043587}{1703178056}a^{14}+\frac{5230829125}{425794514}a^{13}-\frac{36523783249}{1703178056}a^{12}+\frac{6870119456}{212897257}a^{11}-\frac{35946043133}{851589028}a^{10}+\frac{20855645093}{425794514}a^{9}-\frac{10116219730}{212897257}a^{8}+\frac{182356224}{4951099}a^{7}-\frac{4877567924}{212897257}a^{6}+\frac{1917745790}{212897257}a^{5}+\frac{489441230}{212897257}a^{4}-\frac{768417213}{212897257}a^{3}-\frac{17648850}{212897257}a^{2}-\frac{178612744}{212897257}a+\frac{288028651}{212897257}$, $\frac{6497041}{851589028}a^{19}-\frac{234546375}{3406356112}a^{18}+\frac{255214599}{851589028}a^{17}-\frac{1635908977}{1703178056}a^{16}+\frac{4546236739}{1703178056}a^{15}-\frac{10821070041}{1703178056}a^{14}+\frac{11144274101}{851589028}a^{13}-\frac{20633504155}{851589028}a^{12}+\frac{33828661479}{851589028}a^{11}-\frac{24251773445}{425794514}a^{10}+\frac{15464971843}{212897257}a^{9}-\frac{1604127369}{19804396}a^{8}+\frac{32610490325}{425794514}a^{7}-\frac{13077883057}{212897257}a^{6}+\frac{8599325322}{212897257}a^{5}-\frac{7980556693}{425794514}a^{4}+\frac{925570581}{212897257}a^{3}-\frac{305407903}{212897257}a^{2}+\frac{147250678}{212897257}a+\frac{119563402}{212897257}$, $\frac{29913713}{3406356112}a^{19}-\frac{106678021}{1703178056}a^{18}+\frac{201830345}{851589028}a^{17}-\frac{2533545647}{3406356112}a^{16}+\frac{1709001619}{851589028}a^{15}-\frac{1949861267}{425794514}a^{14}+\frac{1992593433}{212897257}a^{13}-\frac{7257101123}{425794514}a^{12}+\frac{23328024079}{851589028}a^{11}-\frac{33666924197}{851589028}a^{10}+\frac{10739406347}{212897257}a^{9}-\frac{24215580171}{425794514}a^{8}+\frac{11980082269}{212897257}a^{7}-\frac{20149585763}{425794514}a^{6}+\frac{7230889038}{212897257}a^{5}-\frac{8044562637}{425794514}a^{4}+\frac{1647458293}{212897257}a^{3}-\frac{392763470}{212897257}a^{2}+\frac{166921140}{212897257}a-\frac{46242785}{212897257}$, $\frac{575489}{425794514}a^{19}-\frac{36374415}{3406356112}a^{18}+\frac{54886575}{851589028}a^{17}-\frac{964346489}{3406356112}a^{16}+\frac{376861705}{425794514}a^{15}-\frac{2022516287}{851589028}a^{14}+\frac{1213770436}{212897257}a^{13}-\frac{4972931699}{425794514}a^{12}+\frac{4559315626}{212897257}a^{11}-\frac{30169244113}{851589028}a^{10}+\frac{10801920339}{212897257}a^{9}-\frac{54926483883}{851589028}a^{8}+\frac{30874139425}{425794514}a^{7}-\frac{29222663297}{425794514}a^{6}+\frac{11718563063}{212897257}a^{5}-\frac{15636599343}{425794514}a^{4}+\frac{3909234751}{212897257}a^{3}-\frac{961776544}{212897257}a^{2}+\frac{297559578}{212897257}a-\frac{317837733}{212897257}$, $\frac{9593707}{3406356112}a^{19}-\frac{74767985}{3406356112}a^{18}+\frac{38822695}{425794514}a^{17}-\frac{128113399}{425794514}a^{16}+\frac{712058627}{851589028}a^{15}-\frac{420880598}{212897257}a^{14}+\frac{7074024745}{1703178056}a^{13}-\frac{13159673149}{1703178056}a^{12}+\frac{2703061429}{212897257}a^{11}-\frac{3959339839}{212897257}a^{10}+\frac{5075320681}{212897257}a^{9}-\frac{5681599914}{212897257}a^{8}+\frac{5471159706}{212897257}a^{7}-\frac{4302805058}{212897257}a^{6}+\frac{2781933731}{212897257}a^{5}-\frac{2588168401}{425794514}a^{4}+\frac{335272120}{212897257}a^{3}-\frac{201312400}{212897257}a^{2}+\frac{195302092}{212897257}a-\frac{94026704}{212897257}$, $\frac{19927393}{851589028}a^{19}-\frac{599847095}{3406356112}a^{18}+\frac{2272295985}{3406356112}a^{17}-\frac{3443373685}{1703178056}a^{16}+\frac{4585865795}{851589028}a^{15}-\frac{2556734359}{212897257}a^{14}+\frac{40041552387}{1703178056}a^{13}-\frac{17647272521}{425794514}a^{12}+\frac{53858530677}{851589028}a^{11}-\frac{71609756885}{851589028}a^{10}+\frac{20979421138}{212897257}a^{9}-\frac{82357465629}{851589028}a^{8}+\frac{32488491025}{425794514}a^{7}-\frac{20129379345}{425794514}a^{6}+\frac{7355659161}{425794514}a^{5}+\frac{2418008309}{425794514}a^{4}-\frac{2049924407}{212897257}a^{3}+\frac{593243675}{212897257}a^{2}-\frac{167020834}{212897257}a+\frac{10336603}{4951099}$, $\frac{18714275}{3406356112}a^{19}-\frac{46633339}{1703178056}a^{18}+\frac{226638001}{3406356112}a^{17}-\frac{294462437}{1703178056}a^{16}+\frac{81410563}{212897257}a^{15}-\frac{932927155}{1703178056}a^{14}+\frac{156897706}{212897257}a^{13}-\frac{550322321}{851589028}a^{12}-\frac{584683439}{851589028}a^{11}+\frac{590783889}{212897257}a^{10}-\frac{5087593149}{851589028}a^{9}+\frac{9127132871}{851589028}a^{8}-\frac{2935475076}{212897257}a^{7}+\frac{6248841069}{425794514}a^{6}-\frac{3036030053}{212897257}a^{5}+\frac{4792982715}{425794514}a^{4}-\frac{1302848272}{212897257}a^{3}+\frac{877472071}{212897257}a^{2}-\frac{666725324}{212897257}a+\frac{125068743}{212897257}$, $\frac{4903759}{1703178056}a^{19}-\frac{94749357}{3406356112}a^{18}+\frac{109460355}{851589028}a^{17}-\frac{1459904783}{3406356112}a^{16}+\frac{2075041921}{1703178056}a^{15}-\frac{5035682599}{1703178056}a^{14}+\frac{1321472845}{212897257}a^{13}-\frac{9939525459}{851589028}a^{12}+\frac{4130359810}{212897257}a^{11}-\frac{24034421167}{851589028}a^{10}+\frac{15489945699}{425794514}a^{9}-\frac{8686458590}{212897257}a^{8}+\frac{16560575149}{425794514}a^{7}-\frac{6761006227}{212897257}a^{6}+\frac{4664073988}{212897257}a^{5}-\frac{2581794536}{212897257}a^{4}+\frac{1275347623}{212897257}a^{3}-\frac{917017896}{212897257}a^{2}+\frac{504314096}{212897257}a-\frac{50759814}{212897257}$, $\frac{13392837}{3406356112}a^{19}-\frac{36028485}{851589028}a^{18}+\frac{329207279}{1703178056}a^{17}-\frac{1061635689}{1703178056}a^{16}+\frac{382798984}{212897257}a^{15}-\frac{1884282107}{425794514}a^{14}+\frac{1989721990}{212897257}a^{13}-\frac{717236501}{39608792}a^{12}+\frac{13259793067}{425794514}a^{11}-\frac{20129362125}{425794514}a^{10}+\frac{14022340189}{212897257}a^{9}-\frac{34676047523}{425794514}a^{8}+\frac{38027176961}{425794514}a^{7}-\frac{443014109}{4951099}a^{6}+\frac{16660510999}{212897257}a^{5}-\frac{25794871757}{425794514}a^{4}+\frac{8887466958}{212897257}a^{3}-\frac{5299661329}{212897257}a^{2}+\frac{2631469664}{212897257}a-\frac{963837694}{212897257}$
| 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: | \( 43044.7776734 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 43044.7776734 \cdot 1}{2\cdot\sqrt{49338146756019243307761664}}\cr\approx \mathstrut & 0.293831086298 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352)
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^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352, 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^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352);
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^20 - 8*x^19 + 34*x^18 - 116*x^17 + 338*x^16 - 836*x^15 + 1840*x^14 - 3640*x^13 + 6408*x^12 - 10176*x^11 + 14560*x^10 - 18560*x^9 + 21128*x^8 - 21328*x^7 + 18704*x^6 - 14048*x^5 + 8976*x^4 - 4576*x^3 + 1760*x^2 - 704*x + 352);
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
$C_2^4:C_5$ (as 20T17):
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 80 |
| The 8 conjugacy class representatives for $C_2^4:C_5$ |
| Character table for $C_2^4:C_5$ |
Intermediate fields
| \(\Q(\zeta_{11})^+\), 10.2.219503494144.1 x2, 10.6.219503494144.2 |
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 10 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | 10.2.219503494144.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.5.0.1}{5} }^{4}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{8}$ | ${\href{/padicField/47.5.0.1}{5} }^{4}$ | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $20$ | $4$ | $5$ | $30$ | |||
|
\(11\)
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
| 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |