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Properties

Label 20.20.6735774334...0000.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 5^{14}\cdot 6029^{9}$
Root discriminant $310.04$
Ramified primes $2, 5, 6029$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7965745361278941149, 0, -1745355717739174201, 0, 146252241763546930, 0, -6421922396464750, 0, 166763466367942, 0, -2699562590419, 0, 27660219998, 0, -176674750, 0, 667370, 0, -1321, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 1321*x^18 + 667370*x^16 - 176674750*x^14 + 27660219998*x^12 - 2699562590419*x^10 + 166763466367942*x^8 - 6421922396464750*x^6 + 146252241763546930*x^4 - 1745355717739174201*x^2 + 7965745361278941149)
 
gp: K = bnfinit(x^20 - 1321*x^18 + 667370*x^16 - 176674750*x^14 + 27660219998*x^12 - 2699562590419*x^10 + 166763466367942*x^8 - 6421922396464750*x^6 + 146252241763546930*x^4 - 1745355717739174201*x^2 + 7965745361278941149, 1)
 

Normalized defining polynomial

\( x^{20} - 1321 x^{18} + 667370 x^{16} - 176674750 x^{14} + 27660219998 x^{12} - 2699562590419 x^{10} + 166763466367942 x^{8} - 6421922396464750 x^{6} + 146252241763546930 x^{4} - 1745355717739174201 x^{2} + 7965745361278941149 \)

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:  \(67357743345283894401969497023167161600000000000000=2^{20}\cdot 5^{14}\cdot 6029^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $310.04$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{6029} a^{10} - \frac{1321}{6029} a^{8} - \frac{1849}{6029} a^{6} - \frac{934}{6029} a^{4}$, $\frac{1}{6029} a^{11} - \frac{1321}{6029} a^{9} - \frac{1849}{6029} a^{7} - \frac{934}{6029} a^{5}$, $\frac{1}{6029} a^{12} + \frac{1520}{6029} a^{8} - \frac{1718}{6029} a^{6} + \frac{2131}{6029} a^{4}$, $\frac{1}{6029} a^{13} + \frac{1520}{6029} a^{9} - \frac{1718}{6029} a^{7} + \frac{2131}{6029} a^{5}$, $\frac{1}{36348841} a^{14} - \frac{1321}{36348841} a^{12} - \frac{1849}{36348841} a^{10} + \frac{16735570}{36348841} a^{8} + \frac{46}{6029} a^{6} + \frac{42}{6029} a^{4}$, $\frac{1}{36348841} a^{15} - \frac{1321}{36348841} a^{13} - \frac{1849}{36348841} a^{11} + \frac{16735570}{36348841} a^{9} + \frac{46}{6029} a^{7} + \frac{42}{6029} a^{5}$, $\frac{1}{7561366432539920667617141425} a^{16} - \frac{97060407763476449329}{7561366432539920667617141425} a^{14} - \frac{531491843153324449547802}{7561366432539920667617141425} a^{12} - \frac{40920933379242508836318}{7561366432539920667617141425} a^{10} + \frac{15097707309462974777700}{50166637469165172782333} a^{8} + \frac{279717864065705452138782}{1254165936729129319558325} a^{6} + \frac{22721087192862585773}{208022215413688724425} a^{4} + \frac{90527120466656946534}{208022215413688724425} a^{2} + \frac{6029}{34503601826785325}$, $\frac{1}{7561366432539920667617141425} a^{17} - \frac{97060407763476449329}{7561366432539920667617141425} a^{15} - \frac{531491843153324449547802}{7561366432539920667617141425} a^{13} - \frac{40920933379242508836318}{7561366432539920667617141425} a^{11} + \frac{15097707309462974777700}{50166637469165172782333} a^{9} + \frac{279717864065705452138782}{1254165936729129319558325} a^{7} + \frac{22721087192862585773}{208022215413688724425} a^{5} + \frac{90527120466656946534}{208022215413688724425} a^{3} + \frac{6029}{34503601826785325} a$, $\frac{1}{42014833141020255538019604968376311075} a^{18} - \frac{2602853259}{42014833141020255538019604968376311075} a^{16} + \frac{127793829901429276983998594393}{42014833141020255538019604968376311075} a^{14} + \frac{1310941855591010713655009721535667}{42014833141020255538019604968376311075} a^{12} - \frac{192707077732536925984706131}{107212149332126148074102364133295} a^{10} + \frac{3208505289467217261907850349560982}{6968789706588199624816653668664175} a^{8} + \frac{52010051498891366525107218783}{1155878206433604183913858628075} a^{6} - \frac{303438103321124710948357183106}{1155878206433604183913858628075} a^{4} + \frac{32952932445350893046255399}{191719722413933352780537175} a^{2} - \frac{3008637816207236116916}{6359917811044397173015}$, $\frac{1}{42014833141020255538019604968376311075} a^{19} - \frac{2602853259}{42014833141020255538019604968376311075} a^{17} + \frac{127793829901429276983998594393}{42014833141020255538019604968376311075} a^{15} + \frac{1310941855591010713655009721535667}{42014833141020255538019604968376311075} a^{13} - \frac{192707077732536925984706131}{107212149332126148074102364133295} a^{11} + \frac{3208505289467217261907850349560982}{6968789706588199624816653668664175} a^{9} + \frac{52010051498891366525107218783}{1155878206433604183913858628075} a^{7} - \frac{303438103321124710948357183106}{1155878206433604183913858628075} a^{5} + \frac{32952932445350893046255399}{191719722413933352780537175} a^{3} - \frac{3008637816207236116916}{6359917811044397173015} a$

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:  \( 2799883032620000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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 $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.2.1$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}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
6029Data not computed