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Properties

Label 20.0.17066738769...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{32}\cdot 31^{18}$
Root discriminant $577.57$
Ramified primes $2, 5, 31$
Class number $1582806000$ (GRH)
Class group $[2, 10, 10, 7914030]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8311744258637824, 0, 2366899073843200, 0, 163111166566400, 0, -5962442238400, 0, 608478061360, 0, -7040729300, 0, 34083105, 0, -1410500, 0, 15190, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 15190*x^16 - 1410500*x^14 + 34083105*x^12 - 7040729300*x^10 + 608478061360*x^8 - 5962442238400*x^6 + 163111166566400*x^4 + 2366899073843200*x^2 + 8311744258637824)
 
gp: K = bnfinit(x^20 + 15190*x^16 - 1410500*x^14 + 34083105*x^12 - 7040729300*x^10 + 608478061360*x^8 - 5962442238400*x^6 + 163111166566400*x^4 + 2366899073843200*x^2 + 8311744258637824, 1)
 

Normalized defining polynomial

\( x^{20} + 15190 x^{16} - 1410500 x^{14} + 34083105 x^{12} - 7040729300 x^{10} + 608478061360 x^{8} - 5962442238400 x^{6} + 163111166566400 x^{4} + 2366899073843200 x^{2} + 8311744258637824 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17066738769507935511210210961914062500000000000000000000=2^{20}\cdot 5^{32}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $577.57$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3100=2^{2}\cdot 5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3100}(1,·)$, $\chi_{3100}(2261,·)$, $\chi_{3100}(771,·)$, $\chi_{3100}(581,·)$, $\chi_{3100}(711,·)$, $\chi_{3100}(2761,·)$, $\chi_{3100}(1551,·)$, $\chi_{3100}(2321,·)$, $\chi_{3100}(2131,·)$, $\chi_{3100}(2581,·)$, $\chi_{3100}(791,·)$, $\chi_{3100}(221,·)$, $\chi_{3100}(1441,·)$, $\chi_{3100}(2851,·)$, $\chi_{3100}(2341,·)$, $\chi_{3100}(1031,·)$, $\chi_{3100}(2991,·)$, $\chi_{3100}(1771,·)$, $\chi_{3100}(1211,·)$, $\chi_{3100}(1301,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} + \frac{1}{32} a^{3} + \frac{1}{8} a$, $\frac{1}{256} a^{8} + \frac{7}{128} a^{6} + \frac{17}{256} a^{4} + \frac{1}{64} a^{2}$, $\frac{1}{256} a^{9} - \frac{1}{128} a^{7} - \frac{15}{256} a^{5} + \frac{13}{64} a^{3} + \frac{1}{4} a$, $\frac{1}{7936} a^{10} - \frac{13}{256} a^{6} + \frac{5}{128} a^{4} + \frac{7}{32} a^{2}$, $\frac{1}{15872} a^{11} - \frac{1}{512} a^{9} + \frac{5}{512} a^{7} + \frac{25}{512} a^{5} + \frac{13}{128} a^{3}$, $\frac{1}{31744} a^{12} - \frac{1}{31744} a^{10} + \frac{1}{1024} a^{8} + \frac{27}{1024} a^{6} - \frac{9}{256} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{63488} a^{13} - \frac{1}{63488} a^{11} + \frac{1}{2048} a^{9} + \frac{27}{2048} a^{7} - \frac{9}{512} a^{5} - \frac{1}{16} a^{3}$, $\frac{1}{63488} a^{14} - \frac{1}{63488} a^{12} - \frac{1}{63488} a^{10} + \frac{3}{2048} a^{8} + \frac{11}{512} a^{6} + \frac{21}{256} a^{4} - \frac{27}{64} a^{2}$, $\frac{1}{507904} a^{15} - \frac{1}{126976} a^{13} + \frac{3}{253952} a^{11} - \frac{1}{4096} a^{9} - \frac{17}{16384} a^{7} + \frac{47}{2048} a^{5} - \frac{81}{1024} a^{3} + \frac{3}{16} a$, $\frac{1}{16252928} a^{16} - \frac{25}{4063232} a^{14} - \frac{77}{8126464} a^{12} - \frac{231}{4063232} a^{10} + \frac{975}{524288} a^{8} - \frac{1137}{65536} a^{6} + \frac{1295}{32768} a^{4} + \frac{59}{512} a^{2} - \frac{1}{2}$, $\frac{1}{32505856} a^{17} + \frac{7}{8126464} a^{15} - \frac{77}{16252928} a^{13} - \frac{167}{8126464} a^{11} + \frac{975}{1048576} a^{9} + \frac{319}{131072} a^{7} + \frac{3151}{65536} a^{5} - \frac{167}{1024} a^{3} - \frac{3}{8} a$, $\frac{1}{53167904909055965935797500039852213972403117621248} a^{18} - \frac{176637385802357200509515188828007994792597}{6645988113631995741974687504981526746550389702656} a^{16} - \frac{99638213535634041578799387277603985137043973}{26583952454527982967898750019926106986201558810624} a^{14} - \frac{325533488750923659412325623087325362308397}{13291976227263991483949375009963053493100779405312} a^{12} - \frac{2066979241296434654572794172879795170783782527}{53167904909055965935797500039852213972403117621248} a^{10} - \frac{91335888262296293130002031953442126604029729}{428773426685935209159657258385904951390347722752} a^{8} + \frac{4092197446502804301398423134205756133363661201}{107193356671483802289914314596476237847586930688} a^{6} + \frac{2454274184602551977936393598911155994388557361}{26798339167870950572478578649119059461896732672} a^{4} + \frac{29151006285811990192721368265549569756582285}{418724049497983602694977791392485304092136448} a^{2} + \frac{97137191165197128207186991647784229626349}{1635640818351498448027256997626895719109908}$, $\frac{1}{4733644909863070759195923023548122314390994368054951936} a^{19} + \frac{126510549249775242024650533333857055369539}{13291976227263991483949375009963053493100779405312} a^{17} - \frac{43105832031547365813232692801956475717984730571}{76349111449404367083805210057227779264370876904112128} a^{15} + \frac{207759244533748181370734592530354416607850030977}{38174555724702183541902605028613889632185438452056064} a^{13} + \frac{557783925505101948922126809052150591602039223103}{152698222898808734167610420114455558528741753808224256} a^{11} - \frac{56094089789881097368425799679473282339845004001795}{38174555724702183541902605028613889632185438452056064} a^{9} - \frac{1904381833978328012172390821443982485747796245997}{307859320360501480176633911521079755098269664935936} a^{7} + \frac{4656986301907738185670230452888988858585160922705}{76964830090125370044158477880269938774567416233984} a^{5} + \frac{140909682071294152210149217297681023474185848725}{1202575470158208906939976216879217793352615878656} a^{3} - \frac{1328717599721116781275555025298273132584360865}{4697560430305503542734282097184444505283655776} a$

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

Class group and class number

$C_{2}\times C_{10}\times C_{10}\times C_{7914030}$, which has order $1582806000$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{591803313435205375}{2960873419691585947763430274367488} a^{19} + \frac{551576372805}{8314070838832065852062826496} a^{17} - \frac{143837896661014691403}{47756022898251386254248875393024} a^{15} + \frac{6754940481642080224545}{23878011449125693127124437696512} a^{13} - \frac{626901794065246530995905}{95512045796502772508497750786048} a^{11} + \frac{32821945244944505853398125}{23878011449125693127124437696512} a^{9} - \frac{23421356084913706910979085}{192564608460691073605842239488} a^{7} + \frac{51853319877981252340943713}{48141152115172768401460559872} a^{5} - \frac{14131218421374694178744895}{752205501799574506272821248} a^{3} - \frac{99388391436217800781355}{183643921337786744695513} a \) (order $4$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 44990028220676.63 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{31}) \), \(\Q(i, \sqrt{31})\), 5.5.360750390625.4, 10.0.133264224600156250000000000.4, 10.0.4034366174418792724609375.1, 10.10.4131190962604843750000000000.2

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$5$5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
5.5.8.2$x^{5} - 5 x^{4} + 5$$5$$1$$8$$C_5$$[2]$
31Data not computed