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Properties

Label 20.4.26512967176...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{22}\cdot 23^{10}$
Root discriminant $74.33$
Ramified primes $2, 5, 23$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![103743262061, 353329680, -4923182660, 11834027480, 13225, 2493672, 153173900, 920, -50590, 701800, -2712032, -880, 11870, -51120, 0, -8, -355, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 355*x^16 - 8*x^15 - 51120*x^13 + 11870*x^12 - 880*x^11 - 2712032*x^10 + 701800*x^9 - 50590*x^8 + 920*x^7 + 153173900*x^6 + 2493672*x^5 + 13225*x^4 + 11834027480*x^3 - 4923182660*x^2 + 353329680*x + 103743262061)
 
gp: K = bnfinit(x^20 - 355*x^16 - 8*x^15 - 51120*x^13 + 11870*x^12 - 880*x^11 - 2712032*x^10 + 701800*x^9 - 50590*x^8 + 920*x^7 + 153173900*x^6 + 2493672*x^5 + 13225*x^4 + 11834027480*x^3 - 4923182660*x^2 + 353329680*x + 103743262061, 1)
 

Normalized defining polynomial

\( x^{20} - 355 x^{16} - 8 x^{15} - 51120 x^{13} + 11870 x^{12} - 880 x^{11} - 2712032 x^{10} + 701800 x^{9} - 50590 x^{8} + 920 x^{7} + 153173900 x^{6} + 2493672 x^{5} + 13225 x^{4} + 11834027480 x^{3} - 4923182660 x^{2} + 353329680 x + 103743262061 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26512967176735360000000000000000000000=2^{28}\cdot 5^{22}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.33$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{5}{12} a + \frac{5}{12}$, $\frac{1}{30972} a^{18} + \frac{721}{30972} a^{17} + \frac{322}{7743} a^{16} - \frac{109}{2581} a^{15} + \frac{539}{5162} a^{14} + \frac{1204}{7743} a^{13} - \frac{626}{7743} a^{12} + \frac{179}{15486} a^{11} - \frac{382}{7743} a^{10} - \frac{2015}{15486} a^{9} - \frac{328}{2581} a^{8} + \frac{3085}{15486} a^{7} - \frac{799}{2581} a^{6} + \frac{6455}{15486} a^{5} - \frac{2521}{5162} a^{4} - \frac{2}{29} a^{3} + \frac{7}{356} a^{2} - \frac{13841}{30972} a + \frac{4289}{15486}$, $\frac{1}{7662846852443524765988784665350737565953148803248521654571835131826195942449348573166913524692652} a^{19} - \frac{8611754632168151254240318260971851673393215639229232763680612242253949583988323716463226874}{1915711713110881191497196166337684391488287200812130413642958782956548985612337143291728381173163} a^{18} + \frac{55446754861475125634353304558211388320250373388333105175774373226037464404350570134187297883339}{3831423426221762382994392332675368782976574401624260827285917565913097971224674286583456762346326} a^{17} - \frac{748647208244999180997382291544345779665059368319217319141383384752630281339727730272871134695975}{7662846852443524765988784665350737565953148803248521654571835131826195942449348573166913524692652} a^{16} + \frac{16441018933614354780915831725805178286185906549690970147589899295603408946137319855588714369787}{2554282284147841588662928221783579188651049601082840551523945043942065314149782857722304508230884} a^{15} + \frac{59072732192162563254803360258610605459860312146405685566650306790499202075616442160639288316550}{1915711713110881191497196166337684391488287200812130413642958782956548985612337143291728381173163} a^{14} - \frac{1360917812652210338114022358254857643028897759289824864699546215654854518526611238244283216128229}{7662846852443524765988784665350737565953148803248521654571835131826195942449348573166913524692652} a^{13} - \frac{261609020825097217079640132454119185838486659828662340477566187470867899785258698544325689885819}{1915711713110881191497196166337684391488287200812130413642958782956548985612337143291728381173163} a^{12} - \frac{55117258866625047631699780779603475074207823937191515071172030608414689705054560758434263765211}{264236098360121543654785678115542674688039613905121436364546039028489515256874088729893569816988} a^{11} - \frac{618200006679016041912195250948085257275158747633321560493903329618044114347572967345195264772655}{3831423426221762382994392332675368782976574401624260827285917565913097971224674286583456762346326} a^{10} - \frac{1273934894675547580032908616472999676745535974427051773098272876523667010709022105578745867606257}{2554282284147841588662928221783579188651049601082840551523945043942065314149782857722304508230884} a^{9} - \frac{317456982759048498016061212327247781479922796904990589741577258191037735486302962893514791591718}{1915711713110881191497196166337684391488287200812130413642958782956548985612337143291728381173163} a^{8} - \frac{1830237929173447926294936659516679648238564867555472051735359720313944740354288989524610819647121}{7662846852443524765988784665350737565953148803248521654571835131826195942449348573166913524692652} a^{7} + \frac{13670721863515503607511327068755308245938978523583448470842119909726396488663716766440552486243}{27966594351983667029156148413688823233405652566600443994787719459219693220618060486010633301798} a^{6} - \frac{2495072573065098099932370329427119878414486655999515264722741533231232864515279822525413779501077}{7662846852443524765988784665350737565953148803248521654571835131826195942449348573166913524692652} a^{5} + \frac{10183910357377106986566621115172777579443910550164740152295237018454370394543894363213805172665}{64939380105453599711769361570768962423331769519055268259083348574798270698723292992939945124514} a^{4} + \frac{851600564313368371976763147144571602390010234208297694687859090845350748386924072394723993750655}{3831423426221762382994392332675368782976574401624260827285917565913097971224674286583456762346326} a^{3} - \frac{2358615774028388148654158613782689631410354738905212332717192805443562556877960748247907966492}{11202992474332638546767229042910435037943200004749300664578706333079233833990275691764493457153} a^{2} + \frac{39325780898389193181186527017269146426603616787848430511635413040344945775560124851116032355345}{129878760210907199423538723141537924846663539038110536518166697149596541397446585985879890249028} a - \frac{3039712946002239661240238393421665854972588860764037266458989740807558431539165966739236238448383}{7662846852443524765988784665350737565953148803248521654571835131826195942449348573166913524692652}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{115}) \), \(\Q(\sqrt{23}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{23})\), 5.1.50000.1, 10.2.1029814880000000000.1, 10.2.5149074400000000000.1, 10.2.12500000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
23.8.4.1$x^{8} + 11638 x^{4} - 12167 x^{2} + 33860761$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$