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Properties

Label 20.12.1753321685...3125.2
Degree $20$
Signature $[12, 4]$
Discriminant $3^{10}\cdot 5^{11}\cdot 239^{10}$
Root discriminant $64.89$
Ramified primes $3, 5, 239$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T426

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1095525, -3782700, -660015, 5565915, 6208866, 2353437, -1925805, -2069502, -237833, 388717, 239526, -62496, -62178, 17195, 7088, -2945, -210, 234, -18, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 - 18*x^18 + 234*x^17 - 210*x^16 - 2945*x^15 + 7088*x^14 + 17195*x^13 - 62178*x^12 - 62496*x^11 + 239526*x^10 + 388717*x^9 - 237833*x^8 - 2069502*x^7 - 1925805*x^6 + 2353437*x^5 + 6208866*x^4 + 5565915*x^3 - 660015*x^2 - 3782700*x - 1095525)
 
gp: K = bnfinit(x^20 - 7*x^19 - 18*x^18 + 234*x^17 - 210*x^16 - 2945*x^15 + 7088*x^14 + 17195*x^13 - 62178*x^12 - 62496*x^11 + 239526*x^10 + 388717*x^9 - 237833*x^8 - 2069502*x^7 - 1925805*x^6 + 2353437*x^5 + 6208866*x^4 + 5565915*x^3 - 660015*x^2 - 3782700*x - 1095525, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} - 18 x^{18} + 234 x^{17} - 210 x^{16} - 2945 x^{15} + 7088 x^{14} + 17195 x^{13} - 62178 x^{12} - 62496 x^{11} + 239526 x^{10} + 388717 x^{9} - 237833 x^{8} - 2069502 x^{7} - 1925805 x^{6} + 2353437 x^{5} + 6208866 x^{4} + 5565915 x^{3} - 660015 x^{2} - 3782700 x - 1095525 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1753321685810638349237472141064453125=3^{10}\cdot 5^{11}\cdot 239^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.89$
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:  $3, 5, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{15} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{2}{9} a^{9} + \frac{1}{3} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{171} a^{17} - \frac{8}{171} a^{16} + \frac{1}{9} a^{15} - \frac{2}{19} a^{14} + \frac{2}{57} a^{13} + \frac{4}{171} a^{12} - \frac{20}{171} a^{11} - \frac{1}{57} a^{10} + \frac{55}{171} a^{9} - \frac{17}{57} a^{8} + \frac{28}{57} a^{7} - \frac{26}{171} a^{6} - \frac{9}{19} a^{5} - \frac{1}{9} a^{4} - \frac{28}{57} a^{3} + \frac{8}{19} a^{2} + \frac{3}{19} a + \frac{8}{19}$, $\frac{1}{2565} a^{18} - \frac{7}{2565} a^{17} + \frac{29}{855} a^{16} + \frac{13}{855} a^{15} + \frac{22}{171} a^{14} + \frac{59}{513} a^{13} + \frac{383}{2565} a^{12} + \frac{22}{513} a^{11} - \frac{46}{855} a^{10} - \frac{347}{855} a^{9} - \frac{22}{95} a^{8} + \frac{457}{2565} a^{7} - \frac{1133}{2565} a^{6} - \frac{41}{95} a^{5} + \frac{59}{171} a^{4} - \frac{52}{285} a^{3} - \frac{46}{285} a^{2} + \frac{2}{19} a - \frac{2}{19}$, $\frac{1}{500302972567860369498120425706324829909982609960061861945} a^{19} + \frac{68661016029973374204362078047577234675868161085568852}{500302972567860369498120425706324829909982609960061861945} a^{18} + \frac{300170265723235304655238129098585405233389098813480949}{500302972567860369498120425706324829909982609960061861945} a^{17} - \frac{6286042754958604459820613700389785641357996263820842956}{166767657522620123166040141902108276636660869986687287315} a^{16} + \frac{2291540670340148827266435528933072950600846399666148142}{15160696138420011196912740172918934239696442726062480665} a^{15} - \frac{13162506770827801081070693951032779738932729292896790268}{100060594513572073899624085141264965981996521992012372389} a^{14} - \frac{3671860119216418461015718283666392054944944137067207972}{500302972567860369498120425706324829909982609960061861945} a^{13} - \frac{7139178118134768829502981195048855439532943219203929316}{166767657522620123166040141902108276636660869986687287315} a^{12} - \frac{69815743379906841913041737280477334750878995852075847083}{500302972567860369498120425706324829909982609960061861945} a^{11} - \frac{5619316645684089120295366764724261354687948700821693832}{55589219174206707722013380634036092212220289995562429105} a^{10} - \frac{22400949637153290357176393090343628695888101750989905602}{55589219174206707722013380634036092212220289995562429105} a^{9} + \frac{138493927714297567609735076098653260742451230526281300046}{500302972567860369498120425706324829909982609960061861945} a^{8} - \frac{3625309108446612452498355893127967419979266598893397417}{33353531504524024633208028380421655327332173997337457463} a^{7} + \frac{139698862659857161184819582794301002651784378267821650731}{500302972567860369498120425706324829909982609960061861945} a^{6} - \frac{4909768775708916228310316699143091942188155977686756734}{18529739724735569240671126878012030737406763331854143035} a^{5} - \frac{71141209754984868149697596293904957220166812441477648071}{166767657522620123166040141902108276636660869986687287315} a^{4} - \frac{3648577815798862612376134345240578960855472248278625763}{18529739724735569240671126878012030737406763331854143035} a^{3} + \frac{4323194608052963666095017624538635652774532061101656182}{18529739724735569240671126878012030737406763331854143035} a^{2} + \frac{114116272904909777011142235168861418900641752121814939}{336904358631555804375838670509309649771032060579166237} a - \frac{103827070047446525207502483778618524995151055149999157}{3705947944947113848134225375602406147481352666370828607}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  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:  \( 195221868018 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T426:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 100 conjugacy class representatives for t20n426 are not computed
Character table for t20n426 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.12852225.1, 10.10.825898437253125.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: 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 $20$ R R $20$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
239Data not computed