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Properties

Label 20.20.7236031593...1744.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{40}\cdot 7^{10}\cdot 13^{12}$
Root discriminant $49.31$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2\times F_5$ (as 20T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 100, 128, -3136, 2795, 17048, -20754, -34168, 46614, 27916, -43824, -10008, 20259, 1484, -4960, -48, 648, -4, -42, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 42*x^18 - 4*x^17 + 648*x^16 - 48*x^15 - 4960*x^14 + 1484*x^13 + 20259*x^12 - 10008*x^11 - 43824*x^10 + 27916*x^9 + 46614*x^8 - 34168*x^7 - 20754*x^6 + 17048*x^5 + 2795*x^4 - 3136*x^3 + 128*x^2 + 100*x + 1)
 
gp: K = bnfinit(x^20 - 42*x^18 - 4*x^17 + 648*x^16 - 48*x^15 - 4960*x^14 + 1484*x^13 + 20259*x^12 - 10008*x^11 - 43824*x^10 + 27916*x^9 + 46614*x^8 - 34168*x^7 - 20754*x^6 + 17048*x^5 + 2795*x^4 - 3136*x^3 + 128*x^2 + 100*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 42 x^{18} - 4 x^{17} + 648 x^{16} - 48 x^{15} - 4960 x^{14} + 1484 x^{13} + 20259 x^{12} - 10008 x^{11} - 43824 x^{10} + 27916 x^{9} + 46614 x^{8} - 34168 x^{7} - 20754 x^{6} + 17048 x^{5} + 2795 x^{4} - 3136 x^{3} + 128 x^{2} + 100 x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7236031593550848872585339280031744=2^{40}\cdot 7^{10}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.31$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1559} a^{18} - \frac{501}{1559} a^{17} - \frac{257}{1559} a^{16} + \frac{502}{1559} a^{15} - \frac{210}{1559} a^{14} - \frac{654}{1559} a^{13} + \frac{340}{1559} a^{12} - \frac{435}{1559} a^{11} + \frac{719}{1559} a^{10} + \frac{111}{1559} a^{9} + \frac{218}{1559} a^{8} + \frac{623}{1559} a^{7} + \frac{544}{1559} a^{6} - \frac{706}{1559} a^{5} - \frac{238}{1559} a^{4} - \frac{486}{1559} a^{3} + \frac{158}{1559} a^{2} - \frac{217}{1559} a - \frac{273}{1559}$, $\frac{1}{1436065374859263057745} a^{19} - \frac{102104471778855759}{1436065374859263057745} a^{18} - \frac{355581429598524862286}{1436065374859263057745} a^{17} + \frac{131100739304927504077}{287213074971852611549} a^{16} - \frac{6611271228260048342}{46324689511589130895} a^{15} - \frac{6434369632806640969}{15116477630097505871} a^{14} - \frac{8141831746521676968}{287213074971852611549} a^{13} + \frac{557261370346191534604}{1436065374859263057745} a^{12} + \frac{321265320687236963438}{1436065374859263057745} a^{11} - \frac{109665430178389725395}{287213074971852611549} a^{10} - \frac{704999100487736860734}{1436065374859263057745} a^{9} + \frac{406988595910295017032}{1436065374859263057745} a^{8} + \frac{132555546708899741571}{1436065374859263057745} a^{7} - \frac{2751125146434690733}{49519495684802174405} a^{6} + \frac{684527494223580876874}{1436065374859263057745} a^{5} - \frac{63984474926983321463}{130551397714478459795} a^{4} + \frac{126232134377422498382}{1436065374859263057745} a^{3} + \frac{10127871614100134789}{49519495684802174405} a^{2} - \frac{139211877353166862591}{1436065374859263057745} a - \frac{666950875473535450916}{1436065374859263057745}$

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:  $19$
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:  \( 73015955617.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times F_5$ (as 20T16):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_2^2\times F_5$
Character table for $C_2^2\times F_5$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{2}, \sqrt{7})\), 5.5.6889792.1, 10.10.379753870426112.1, 10.10.42532433487724544.1, 10.10.21266216743862272.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$