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Properties

Label 20.12.1136318612...0000.1
Degree $20$
Signature $[12, 4]$
Discriminant $2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{10}\cdot 31^{4}$
Root discriminant $79.94$
Ramified primes $2, 5, 13, 29, 31$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-56509, 5360629, 136837803, 121164707, 44334749, -40761296, -60534112, 3352516, 17793574, -233250, -2192766, 80254, 85642, -4704, 3960, -676, -177, 61, -21, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 21*x^18 + 61*x^17 - 177*x^16 - 676*x^15 + 3960*x^14 - 4704*x^13 + 85642*x^12 + 80254*x^11 - 2192766*x^10 - 233250*x^9 + 17793574*x^8 + 3352516*x^7 - 60534112*x^6 - 40761296*x^5 + 44334749*x^4 + 121164707*x^3 + 136837803*x^2 + 5360629*x - 56509)
 
gp: K = bnfinit(x^20 - x^19 - 21*x^18 + 61*x^17 - 177*x^16 - 676*x^15 + 3960*x^14 - 4704*x^13 + 85642*x^12 + 80254*x^11 - 2192766*x^10 - 233250*x^9 + 17793574*x^8 + 3352516*x^7 - 60534112*x^6 - 40761296*x^5 + 44334749*x^4 + 121164707*x^3 + 136837803*x^2 + 5360629*x - 56509, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 21 x^{18} + 61 x^{17} - 177 x^{16} - 676 x^{15} + 3960 x^{14} - 4704 x^{13} + 85642 x^{12} + 80254 x^{11} - 2192766 x^{10} - 233250 x^{9} + 17793574 x^{8} + 3352516 x^{7} - 60534112 x^{6} - 40761296 x^{5} + 44334749 x^{4} + 121164707 x^{3} + 136837803 x^{2} + 5360629 x - 56509 \)

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:  \(113631861215080224467348285440000000000=2^{20}\cdot 5^{10}\cdot 13^{4}\cdot 29^{10}\cdot 31^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.94$
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:  $2, 5, 13, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{9} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{2} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{14} + \frac{1}{8} a^{11} - \frac{1}{4} a^{10} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{3}{8} a^{6} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{17} - \frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{3}{8} a^{5} - \frac{3}{8} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{8} a^{18} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{19} + \frac{309206309215018764015127136843797801805203527406772592259293322737301}{5691336861130206545672660975378804594759238258210369014284979997463096} a^{18} - \frac{408085343689173390092588873410950353933258254600118234872267884794239}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{17} - \frac{48922761718730314591245431793845388952413844488638437752506663658435}{5691336861130206545672660975378804594759238258210369014284979997463096} a^{16} + \frac{368433577993927031575721802389262441718054629820796022564394249384367}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{15} - \frac{548115017957130305985970153718556616126180859901397263516034316369485}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{14} - \frac{507959012025655890199556493771079072481120020685548905667580447469625}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{13} + \frac{1273952833964959901025039070210175769549410173242274142579647607512197}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{12} + \frac{464250798663686198770540568417138528322116616699341235928514739395719}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{11} + \frac{1997788294282413688609897773145336134708340711657671267480306139518789}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{10} + \frac{432724292328784372322220648763499183053236570212945174598367143936463}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{9} + \frac{2271592801899436279419232607246852792925182583225480224179907951395091}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{8} - \frac{2428593836313230365564295803941428408211516625069138551156570560312931}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{7} + \frac{2003318382960737102957251925242522368419890643912299674706507585731481}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{6} + \frac{2840748136533680481044212010181582170727898933370892674881069954460133}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{5} - \frac{4760634258862329367859438644411497605604568368549603470040621168077569}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{4} - \frac{297830463414974646458100860446117459464575265515041648056239476519837}{2845668430565103272836330487689402297379619129105184507142489998731548} a^{3} + \frac{4424881739205442145708149346369061201506474286320472919147283120544245}{11382673722260413091345321950757609189518476516420738028569959994926192} a^{2} + \frac{429821019348122653438931448286642877465541439204583949539188140026881}{2845668430565103272836330487689402297379619129105184507142489998731548} a - \frac{2649550584881544086142401123670742810877520450432907581243644535373585}{11382673722260413091345321950757609189518476516420738028569959994926192}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 463600803485 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R R ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
2Data not computed
5Data not computed
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
29.10.5.1$x^{10} - 1682 x^{6} + 707281 x^{2} - 2481849029$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$31$$\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$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.6.4.3$x^{6} + 713 x^{3} + 138384$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.8.0.1$x^{8} - x + 22$$1$$8$$0$$C_8$$[\ ]^{8}$