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Properties

Label 20.16.8983365148...0000.1
Degree $20$
Signature $[16, 2]$
Discriminant $2^{20}\cdot 5^{10}\cdot 41^{5}\cdot 27517559^{2}$
Root discriminant $62.76$
Ramified primes $2, 5, 41, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T781

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11181241, 50121528, -41887024, -98760184, 48979284, 75872184, -28527308, -30275176, 9888373, 7098988, -2182234, -1048588, 313830, 101334, -29556, -6364, 1781, 238, -63, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 63*x^18 + 238*x^17 + 1781*x^16 - 6364*x^15 - 29556*x^14 + 101334*x^13 + 313830*x^12 - 1048588*x^11 - 2182234*x^10 + 7098988*x^9 + 9888373*x^8 - 30275176*x^7 - 28527308*x^6 + 75872184*x^5 + 48979284*x^4 - 98760184*x^3 - 41887024*x^2 + 50121528*x + 11181241)
 
gp: K = bnfinit(x^20 - 4*x^19 - 63*x^18 + 238*x^17 + 1781*x^16 - 6364*x^15 - 29556*x^14 + 101334*x^13 + 313830*x^12 - 1048588*x^11 - 2182234*x^10 + 7098988*x^9 + 9888373*x^8 - 30275176*x^7 - 28527308*x^6 + 75872184*x^5 + 48979284*x^4 - 98760184*x^3 - 41887024*x^2 + 50121528*x + 11181241, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 63 x^{18} + 238 x^{17} + 1781 x^{16} - 6364 x^{15} - 29556 x^{14} + 101334 x^{13} + 313830 x^{12} - 1048588 x^{11} - 2182234 x^{10} + 7098988 x^{9} + 9888373 x^{8} - 30275176 x^{7} - 28527308 x^{6} + 75872184 x^{5} + 48979284 x^{4} - 98760184 x^{3} - 41887024 x^{2} + 50121528 x + 11181241 \)

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:  \(898336514802612753926973440000000000=2^{20}\cdot 5^{10}\cdot 41^{5}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.76$
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:  $2, 5, 41, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{10313649147140794392780378801030876507445977832587252079936867849} a^{19} - \frac{5155678379496342101233095995422015180430453292909814903921915672}{10313649147140794392780378801030876507445977832587252079936867849} a^{18} - \frac{29149306236434063367897662580314242007160133944812011555160030}{937604467921890399343670800093716046131452530235204734539715259} a^{17} + \frac{4800591709879886643442632000007724580236436517668937957183196672}{10313649147140794392780378801030876507445977832587252079936867849} a^{16} + \frac{2311279308270644257608767342462778992596728244711800262732189926}{10313649147140794392780378801030876507445977832587252079936867849} a^{15} - \frac{199052320592188846790968573751214940530395396996433432915400255}{937604467921890399343670800093716046131452530235204734539715259} a^{14} - \frac{285929977097835776362766574044594655145388929737443103252877833}{10313649147140794392780378801030876507445977832587252079936867849} a^{13} + \frac{2192434332718009165227615573749137805982331660340183623077619217}{10313649147140794392780378801030876507445977832587252079936867849} a^{12} - \frac{3261207852687554816814186083975429290696356938891362113206979657}{10313649147140794392780378801030876507445977832587252079936867849} a^{11} + \frac{5014853044716980757021322633085390653159622088695998892916762503}{10313649147140794392780378801030876507445977832587252079936867849} a^{10} - \frac{519816361525531178951242552218024384662829032867304432349142269}{10313649147140794392780378801030876507445977832587252079936867849} a^{9} + \frac{1597007960525752128199756543515552924534373594408096907011259622}{10313649147140794392780378801030876507445977832587252079936867849} a^{8} - \frac{1122619366239784475151676404781118428832937847335018748847350256}{10313649147140794392780378801030876507445977832587252079936867849} a^{7} + \frac{445530301326341694132278788921616952058956963795377946157007275}{937604467921890399343670800093716046131452530235204734539715259} a^{6} - \frac{4870174270239249727654747165437792320304507284879306719166877169}{10313649147140794392780378801030876507445977832587252079936867849} a^{5} - \frac{4848521209745770477586599967074714272781087574500470694318894669}{10313649147140794392780378801030876507445977832587252079936867849} a^{4} - \frac{3376408629057469108914390900993036373213589563978696206991980939}{10313649147140794392780378801030876507445977832587252079936867849} a^{3} - \frac{2526920408866625595397348520621408669041034888831167615149593873}{10313649147140794392780378801030876507445977832587252079936867849} a^{2} - \frac{44883525497377163431144739926944559843896369701431248514062351}{937604467921890399343670800093716046131452530235204734539715259} a + \frac{683963046496756278689047300465209065412446451452376739319881377}{10313649147140794392780378801030876507445977832587252079936867849}$

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:  \( 115508854851 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T781:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 115200
The 119 conjugacy class representatives for t20n781 are not computed
Character table for t20n781 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.16400.1, 10.8.85992371875.1

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 24 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 R $20$ R $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.8.4.1$x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
27517559Data not computed