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Properties

Label 20.4.31499904694...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{16}\cdot 5^{22}\cdot 17^{10}$
Root discriminant $42.16$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9130279, 1555120, -14624315, -542730, 10342455, -91742, -4465140, 71690, 1364105, -14970, -313455, 1830, 54495, -110, -6940, -4, 615, 0, -35, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 35*x^18 + 615*x^16 - 4*x^15 - 6940*x^14 - 110*x^13 + 54495*x^12 + 1830*x^11 - 313455*x^10 - 14970*x^9 + 1364105*x^8 + 71690*x^7 - 4465140*x^6 - 91742*x^5 + 10342455*x^4 - 542730*x^3 - 14624315*x^2 + 1555120*x + 9130279)
 
gp: K = bnfinit(x^20 - 35*x^18 + 615*x^16 - 4*x^15 - 6940*x^14 - 110*x^13 + 54495*x^12 + 1830*x^11 - 313455*x^10 - 14970*x^9 + 1364105*x^8 + 71690*x^7 - 4465140*x^6 - 91742*x^5 + 10342455*x^4 - 542730*x^3 - 14624315*x^2 + 1555120*x + 9130279, 1)
 

Normalized defining polynomial

\( x^{20} - 35 x^{18} + 615 x^{16} - 4 x^{15} - 6940 x^{14} - 110 x^{13} + 54495 x^{12} + 1830 x^{11} - 313455 x^{10} - 14970 x^{9} + 1364105 x^{8} + 71690 x^{7} - 4465140 x^{6} - 91742 x^{5} + 10342455 x^{4} - 542730 x^{3} - 14624315 x^{2} + 1555120 x + 9130279 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(314999046945156250000000000000000=2^{16}\cdot 5^{22}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.16$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2114} a^{18} - \frac{47}{1057} a^{17} - \frac{185}{2114} a^{16} - \frac{219}{1057} a^{15} + \frac{313}{2114} a^{14} - \frac{115}{1057} a^{13} - \frac{59}{2114} a^{12} + \frac{248}{1057} a^{11} + \frac{16}{1057} a^{10} - \frac{865}{2114} a^{9} + \frac{347}{2114} a^{8} + \frac{373}{1057} a^{7} + \frac{221}{2114} a^{6} - \frac{939}{2114} a^{5} + \frac{180}{1057} a^{4} + \frac{12}{1057} a^{3} + \frac{85}{2114} a^{2} - \frac{59}{2114} a - \frac{29}{1057}$, $\frac{1}{122772912320350325903518215430415537138454302558330802} a^{19} - \frac{18566127876464326569231618589356653781133327061309}{122772912320350325903518215430415537138454302558330802} a^{18} - \frac{12003044707756693297672074350250905451163404002062285}{122772912320350325903518215430415537138454302558330802} a^{17} - \frac{16499284391151368690019469007621490639977623188033457}{122772912320350325903518215430415537138454302558330802} a^{16} + \frac{3550929410936178482962595971043343521316443151731939}{122772912320350325903518215430415537138454302558330802} a^{15} + \frac{15117722064641440411621012134582462038029994109502571}{61386456160175162951759107715207768569227151279165401} a^{14} + \frac{7181552700684671541945707947187583917656501825025127}{40924304106783441967839405143471845712818100852776934} a^{13} + \frac{1227038154017431826578586819005505610449481329168768}{8769493737167880421679872530743966938461021611309343} a^{12} - \frac{13870856848050130498244921947261971809087017997633209}{61386456160175162951759107715207768569227151279165401} a^{11} + \frac{11985920483324548904997170011770840273835768099379381}{61386456160175162951759107715207768569227151279165401} a^{10} + \frac{44096224512640769710189404117227400728920441933130791}{122772912320350325903518215430415537138454302558330802} a^{9} + \frac{51787357047723555522249507287100274798615625843566639}{122772912320350325903518215430415537138454302558330802} a^{8} + \frac{3992805374421289370827855331299479686135824028173579}{40924304106783441967839405143471845712818100852776934} a^{7} + \frac{3815199295799468694084104884401123940278819986940371}{17538987474335760843359745061487933876922043222618686} a^{6} - \frac{11653367382453466910294605821659286651298340958696509}{61386456160175162951759107715207768569227151279165401} a^{5} - \frac{6283390743295069593077942397882447515445323137380253}{20462152053391720983919702571735922856409050426388467} a^{4} - \frac{2818432664194595356875492941038876913480176250736917}{5846329158111920281119915020495977958974014407539562} a^{3} + \frac{8647786877541814540430352426540318062465291691418292}{20462152053391720983919702571735922856409050426388467} a^{2} - \frac{28284090530550863531404004954299236475837730667044563}{61386456160175162951759107715207768569227151279165401} a - \frac{21905958744030298179048596314286055255951484990569295}{122772912320350325903518215430415537138454302558330802}$

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:  $11$
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:  \( 420889467.5752704 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T13):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{85}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{17})\), 5.1.50000.1, 10.2.3549642500000000.1, 10.2.17748212500000000.1, 10.2.12500000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$2$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$