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Properties

Label 20.16.9801220092...7936.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{52}\cdot 31^{10}\cdot 227^{4}$
Root discriminant $99.90$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2125814384, 10166002240, -492654464, -9649067552, 468605560, 2477753184, 46964488, -207011648, -57112310, -7077232, 9178424, 2196536, -431938, -134036, -17904, 3304, 2486, -28, -86, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 86*x^18 - 28*x^17 + 2486*x^16 + 3304*x^15 - 17904*x^14 - 134036*x^13 - 431938*x^12 + 2196536*x^11 + 9178424*x^10 - 7077232*x^9 - 57112310*x^8 - 207011648*x^7 + 46964488*x^6 + 2477753184*x^5 + 468605560*x^4 - 9649067552*x^3 - 492654464*x^2 + 10166002240*x + 2125814384)
 
gp: K = bnfinit(x^20 - 86*x^18 - 28*x^17 + 2486*x^16 + 3304*x^15 - 17904*x^14 - 134036*x^13 - 431938*x^12 + 2196536*x^11 + 9178424*x^10 - 7077232*x^9 - 57112310*x^8 - 207011648*x^7 + 46964488*x^6 + 2477753184*x^5 + 468605560*x^4 - 9649067552*x^3 - 492654464*x^2 + 10166002240*x + 2125814384, 1)
 

Normalized defining polynomial

\( x^{20} - 86 x^{18} - 28 x^{17} + 2486 x^{16} + 3304 x^{15} - 17904 x^{14} - 134036 x^{13} - 431938 x^{12} + 2196536 x^{11} + 9178424 x^{10} - 7077232 x^{9} - 57112310 x^{8} - 207011648 x^{7} + 46964488 x^{6} + 2477753184 x^{5} + 468605560 x^{4} - 9649067552 x^{3} - 492654464 x^{2} + 10166002240 x + 2125814384 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(9801220092153173931756020237102466727936=2^{52}\cdot 31^{10}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.90$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{62} a^{12} + \frac{15}{31} a^{11} - \frac{3}{31} a^{10} + \frac{2}{31} a^{9} + \frac{5}{31} a^{8} + \frac{6}{31} a^{7} - \frac{5}{31} a^{6} - \frac{14}{31} a^{5} + \frac{7}{31} a^{4} - \frac{2}{31} a^{3} - \frac{2}{31} a^{2} + \frac{8}{31} a + \frac{12}{31}$, $\frac{1}{62} a^{13} + \frac{12}{31} a^{11} - \frac{1}{31} a^{10} + \frac{7}{31} a^{9} + \frac{11}{31} a^{8} + \frac{1}{31} a^{7} + \frac{12}{31} a^{6} - \frac{7}{31} a^{5} + \frac{5}{31} a^{4} - \frac{4}{31} a^{3} + \frac{6}{31} a^{2} - \frac{11}{31} a + \frac{12}{31}$, $\frac{1}{1922} a^{14} + \frac{7}{1922} a^{13} + \frac{1}{961} a^{12} - \frac{278}{961} a^{11} + \frac{128}{961} a^{10} - \frac{387}{961} a^{9} - \frac{404}{961} a^{8} + \frac{290}{961} a^{7} - \frac{216}{961} a^{6} - \frac{108}{961} a^{5} + \frac{342}{961} a^{4} - \frac{288}{961} a^{3} - \frac{328}{961} a^{2} - \frac{148}{961} a - \frac{366}{961}$, $\frac{1}{3844} a^{15} - \frac{4}{961} a^{13} - \frac{3}{961} a^{12} + \frac{245}{1922} a^{11} + \frac{428}{961} a^{10} - \frac{103}{961} a^{9} - \frac{239}{961} a^{8} - \frac{789}{1922} a^{7} + \frac{454}{961} a^{6} - \frac{102}{961} a^{5} + \frac{209}{961} a^{4} - \frac{513}{1922} a^{3} - \frac{352}{961} a^{2} - \frac{6}{961} a + \frac{10}{961}$, $\frac{1}{119164} a^{16} + \frac{15}{119164} a^{15} + \frac{3}{29791} a^{14} - \frac{431}{59582} a^{13} - \frac{110}{29791} a^{12} - \frac{29}{59582} a^{11} - \frac{9499}{29791} a^{10} + \frac{4547}{29791} a^{9} + \frac{507}{59582} a^{8} + \frac{28069}{59582} a^{7} - \frac{7786}{29791} a^{6} + \frac{1786}{29791} a^{5} - \frac{3949}{59582} a^{4} + \frac{24891}{59582} a^{3} + \frac{9652}{29791} a^{2} + \frac{12511}{29791} a - \frac{12724}{29791}$, $\frac{1}{119164} a^{17} + \frac{1}{29791} a^{15} + \frac{3}{29791} a^{14} - \frac{451}{59582} a^{13} + \frac{70}{29791} a^{12} - \frac{8119}{29791} a^{11} - \frac{3039}{29791} a^{10} - \frac{6137}{59582} a^{9} - \frac{6601}{29791} a^{8} + \frac{1719}{29791} a^{7} + \frac{13889}{29791} a^{6} - \frac{13819}{59582} a^{5} - \frac{4313}{29791} a^{4} + \frac{13387}{29791} a^{3} - \frac{10563}{29791} a^{2} + \frac{925}{29791} a + \frac{10719}{29791}$, $\frac{1}{7388168} a^{18} - \frac{1}{923521} a^{17} + \frac{3}{923521} a^{16} - \frac{147}{1847042} a^{15} + \frac{303}{3694084} a^{14} - \frac{6187}{1847042} a^{13} - \frac{433}{923521} a^{12} - \frac{263133}{1847042} a^{11} + \frac{1077615}{3694084} a^{10} + \frac{315541}{923521} a^{9} - \frac{93047}{1847042} a^{8} + \frac{68881}{923521} a^{7} - \frac{319447}{3694084} a^{6} - \frac{110186}{923521} a^{5} - \frac{41}{59582} a^{4} + \frac{327157}{923521} a^{3} + \frac{213309}{923521} a^{2} + \frac{342700}{923521} a - \frac{304067}{923521}$, $\frac{1}{296025149188340833527436008200482771318434264854700659064} a^{19} - \frac{1132023960019171501106675052082107000299741470705}{74006287297085208381859002050120692829608566213675164766} a^{18} + \frac{409729514399402583922967497956267763767040064811751}{148012574594170416763718004100241385659217132427350329532} a^{17} - \frac{152983074892691791490115139293643749375477663435105}{49337524864723472254572668033413795219739044142450109844} a^{16} - \frac{6826652090327742133975616294930099610460824730592113}{74006287297085208381859002050120692829608566213675164766} a^{15} - \frac{295083237970042880912584455272510022231573556829543}{1193649795114277554546112936292269239187234938930244593} a^{14} + \frac{405485596629344072436153106014361031396103720144872641}{74006287297085208381859002050120692829608566213675164766} a^{13} - \frac{64996290686718141302167955539811729503773150354596013}{24668762432361736127286334016706897609869522071225054922} a^{12} - \frac{66025851714523886217939805880210518556388703481483409443}{148012574594170416763718004100241385659217132427350329532} a^{11} - \frac{3372643603609631922522908046595158591050063860683944794}{12334381216180868063643167008353448804934761035612527461} a^{10} - \frac{13256980361870876112209777666426875318953939505619345540}{37003143648542604190929501025060346414804283106837582383} a^{9} + \frac{11742064649497687001453387817826938846051934534623337363}{24668762432361736127286334016706897609869522071225054922} a^{8} + \frac{66781681560208626661090903102125895492456568744350258103}{148012574594170416763718004100241385659217132427350329532} a^{7} + \frac{959305811939162506515047259309953322077892445672952048}{12334381216180868063643167008353448804934761035612527461} a^{6} + \frac{12782929203947804200188358755544984962183047940824482271}{37003143648542604190929501025060346414804283106837582383} a^{5} + \frac{26636199883747854638124255322019971686507582814396995035}{74006287297085208381859002050120692829608566213675164766} a^{4} - \frac{14346975562084552241806854573051682985185367303501503693}{74006287297085208381859002050120692829608566213675164766} a^{3} + \frac{7724650542256814431607234900689786942826718667000251439}{37003143648542604190929501025060346414804283106837582383} a^{2} - \frac{16587132132695245045822415300491031381045566067562103868}{37003143648542604190929501025060346414804283106837582383} a + \frac{12925410468972093555672261130788990348588185575567838779}{37003143648542604190929501025060346414804283106837582383}$

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:  $17$
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:  \( 63261410870600 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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

Sibling fields

Degree 20 sibling: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.8.0.1}{8} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{3}$

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$2.8.24.7$x^{8} + 8 x^{7} + 12 x^{6} + 10 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 14$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.12.28.195$x^{12} + 2 x^{10} + 2 x^{8} + 4 x^{5} + 2 x^{4} - 2$$12$$1$$28$12T48$[2, 8/3, 8/3, 3]_{3}^{2}$
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.6.5.5$x^{6} + 10633$$6$$1$$5$$C_6$$[\ ]_{6}$
31.10.5.1$x^{10} - 1922 x^{6} + 923521 x^{2} - 2862915100$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
227Data not computed