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Properties

Label 20.8.11863344525...0000.2
Degree $20$
Signature $[8, 6]$
Discriminant $2^{24}\cdot 5^{11}\cdot 3469^{4}$
Root discriminant $28.43$
Ramified primes $2, 5, 3469$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T755

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 864, 1757, -5366, 4385, 592, -5680, 4316, 401, -2914, 1845, 336, -979, 366, 150, -180, 25, 32, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 11*x^18 + 32*x^17 + 25*x^16 - 180*x^15 + 150*x^14 + 366*x^13 - 979*x^12 + 336*x^11 + 1845*x^10 - 2914*x^9 + 401*x^8 + 4316*x^7 - 5680*x^6 + 592*x^5 + 4385*x^4 - 5366*x^3 + 1757*x^2 + 864*x + 31)
 
gp: K = bnfinit(x^20 - 2*x^19 - 11*x^18 + 32*x^17 + 25*x^16 - 180*x^15 + 150*x^14 + 366*x^13 - 979*x^12 + 336*x^11 + 1845*x^10 - 2914*x^9 + 401*x^8 + 4316*x^7 - 5680*x^6 + 592*x^5 + 4385*x^4 - 5366*x^3 + 1757*x^2 + 864*x + 31, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 11 x^{18} + 32 x^{17} + 25 x^{16} - 180 x^{15} + 150 x^{14} + 366 x^{13} - 979 x^{12} + 336 x^{11} + 1845 x^{10} - 2914 x^{9} + 401 x^{8} + 4316 x^{7} - 5680 x^{6} + 592 x^{5} + 4385 x^{4} - 5366 x^{3} + 1757 x^{2} + 864 x + 31 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(118633445252727603200000000000=2^{24}\cdot 5^{11}\cdot 3469^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.43$
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:  $2, 5, 3469$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{851168216205355757940432786077} a^{19} + \frac{409513324174147784370884373411}{851168216205355757940432786077} a^{18} - \frac{351468977915913829895709033449}{851168216205355757940432786077} a^{17} - \frac{401056175339490034691521018083}{851168216205355757940432786077} a^{16} + \frac{279208557306792966268139527674}{851168216205355757940432786077} a^{15} - \frac{408929746978374817712624682890}{851168216205355757940432786077} a^{14} + \frac{425453673834399677015945031577}{851168216205355757940432786077} a^{13} - \frac{302629061749133915689250544076}{851168216205355757940432786077} a^{12} + \frac{15358684767202569653597361013}{851168216205355757940432786077} a^{11} + \frac{400872264842678616119449935630}{851168216205355757940432786077} a^{10} + \frac{53269894355767193520331386960}{851168216205355757940432786077} a^{9} + \frac{391269556677629241119680699678}{851168216205355757940432786077} a^{8} - \frac{229471223011585308881606555922}{851168216205355757940432786077} a^{7} + \frac{47920477585138741889205991900}{851168216205355757940432786077} a^{6} - \frac{89131820531301603141044263803}{851168216205355757940432786077} a^{5} - \frac{19642841082989417903568856029}{851168216205355757940432786077} a^{4} - \frac{203723180259800648482908326816}{851168216205355757940432786077} a^{3} - \frac{368403147514851969590547711261}{851168216205355757940432786077} a^{2} - \frac{88130368955769910190095141415}{851168216205355757940432786077} a + \frac{184208110548106934582155001307}{851168216205355757940432786077}$

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:  $13$
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:  \( 11393335.2276 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T755:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n755 are not computed
Character table for t20n755 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed