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Properties

Label 20.6.14113234290...1603.1
Degree $20$
Signature $[6, 7]$
Discriminant $-\,13^{12}\cdot 347^{7}$
Root discriminant $36.10$
Ramified primes $13, 347$
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![13913, 57338, 16845, -29505, 141149, 60723, -63278, 116976, 8298, 3687, 23782, -6403, -1961, 993, -888, -27, 70, -46, 19, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 + 19*x^18 - 46*x^17 + 70*x^16 - 27*x^15 - 888*x^14 + 993*x^13 - 1961*x^12 - 6403*x^11 + 23782*x^10 + 3687*x^9 + 8298*x^8 + 116976*x^7 - 63278*x^6 + 60723*x^5 + 141149*x^4 - 29505*x^3 + 16845*x^2 + 57338*x + 13913)
 
gp: K = bnfinit(x^20 - 3*x^19 + 19*x^18 - 46*x^17 + 70*x^16 - 27*x^15 - 888*x^14 + 993*x^13 - 1961*x^12 - 6403*x^11 + 23782*x^10 + 3687*x^9 + 8298*x^8 + 116976*x^7 - 63278*x^6 + 60723*x^5 + 141149*x^4 - 29505*x^3 + 16845*x^2 + 57338*x + 13913, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} + 19 x^{18} - 46 x^{17} + 70 x^{16} - 27 x^{15} - 888 x^{14} + 993 x^{13} - 1961 x^{12} - 6403 x^{11} + 23782 x^{10} + 3687 x^{9} + 8298 x^{8} + 116976 x^{7} - 63278 x^{6} + 60723 x^{5} + 141149 x^{4} - 29505 x^{3} + 16845 x^{2} + 57338 x + 13913 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-14113234290632913459536084681603=-\,13^{12}\cdot 347^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.10$
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:  $13, 347$
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}{241460541834369567919315757780866042882041828436784876183} a^{19} - \frac{14340624854511447223801197190757427191298454118221209684}{241460541834369567919315757780866042882041828436784876183} a^{18} - \frac{37855185719802105849370877992514720146132567405835279147}{241460541834369567919315757780866042882041828436784876183} a^{17} - \frac{12816045317618060913432339520846521638614332016119832874}{241460541834369567919315757780866042882041828436784876183} a^{16} + \frac{9561878133740869155625321592609567196177739265214556333}{241460541834369567919315757780866042882041828436784876183} a^{15} + \frac{63894427387094451519608851914137739706590129647534683170}{241460541834369567919315757780866042882041828436784876183} a^{14} + \frac{97432318695823689999764311967535461143893823650567638806}{241460541834369567919315757780866042882041828436784876183} a^{13} - \frac{26374784382201904604475163659676810312555885891652420532}{241460541834369567919315757780866042882041828436784876183} a^{12} - \frac{43339128210856036687717720291160143527661694589038750881}{241460541834369567919315757780866042882041828436784876183} a^{11} - \frac{50247634848701648951353604045575960300102919760408626199}{241460541834369567919315757780866042882041828436784876183} a^{10} + \frac{18951768101673694976275176993813270792793173646457155554}{241460541834369567919315757780866042882041828436784876183} a^{9} + \frac{54675840284940827367027983471850197005474345215720710229}{241460541834369567919315757780866042882041828436784876183} a^{8} - \frac{86087960215670042547699544250061564705583866053468300668}{241460541834369567919315757780866042882041828436784876183} a^{7} + \frac{71010116927940094772731753673770578754534213995284241147}{241460541834369567919315757780866042882041828436784876183} a^{6} + \frac{39721097022528921538069887800592792574484218561477701858}{241460541834369567919315757780866042882041828436784876183} a^{5} + \frac{111915977532466458309707382720965214470258458515703711845}{241460541834369567919315757780866042882041828436784876183} a^{4} - \frac{68165682464982775055266664367873271468561494436122818804}{241460541834369567919315757780866042882041828436784876183} a^{3} + \frac{8262826093594125338816968720019957299985353233177087570}{241460541834369567919315757780866042882041828436784876183} a^{2} + \frac{94814717301785571441318969079326815870710383356274514064}{241460541834369567919315757780866042882041828436784876183} a - \frac{102285346920851734150333770113377118320915819984225882798}{241460541834369567919315757780866042882041828436784876183}$

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:  $12$
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:  \( 48426682.1436 \) (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

\(\Q(\sqrt{13}) \), 5.3.4511.1, 10.6.44707018837.1

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 $20$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ R ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.8.6.2$x^{8} + 39 x^{4} + 676$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
347Data not computed