↑

Properties

Label 28.0.17249157379...0000.2
Degree $28$
Signature $[0, 14]$
Discriminant $2^{28}\cdot 5^{14}\cdot 29^{26}$
Root discriminant $101.97$
Ramified primes $2, 5, 29$
Class number $311808$ (GRH)
Class group $[2, 4, 4, 4, 2436]$ (GRH)
Galois group $C_2\times C_{14}$ (as 28T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7123561, 0, 89938214, 0, 350752457, 0, 692717515, 0, 820109500, 0, 629891623, 0, 327131195, 0, 117391053, 0, 29379286, 0, 5126412, 0, 617080, 0, 49961, 0, 2586, 0, 77, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^28 + 77*x^26 + 2586*x^24 + 49961*x^22 + 617080*x^20 + 5126412*x^18 + 29379286*x^16 + 117391053*x^14 + 327131195*x^12 + 629891623*x^10 + 820109500*x^8 + 692717515*x^6 + 350752457*x^4 + 89938214*x^2 + 7123561)
 
gp: K = bnfinit(x^28 + 77*x^26 + 2586*x^24 + 49961*x^22 + 617080*x^20 + 5126412*x^18 + 29379286*x^16 + 117391053*x^14 + 327131195*x^12 + 629891623*x^10 + 820109500*x^8 + 692717515*x^6 + 350752457*x^4 + 89938214*x^2 + 7123561, 1)
 

Normalized defining polynomial

\( x^{28} + 77 x^{26} + 2586 x^{24} + 49961 x^{22} + 617080 x^{20} + 5126412 x^{18} + 29379286 x^{16} + 117391053 x^{14} + 327131195 x^{12} + 629891623 x^{10} + 820109500 x^{8} + 692717515 x^{6} + 350752457 x^{4} + 89938214 x^{2} + 7123561 \)

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

Invariants

Degree:  $28$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 14]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(172491573797176117219488267129250761475686400000000000000=2^{28}\cdot 5^{14}\cdot 29^{26}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.97$
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:  $2, 5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(580=2^{2}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{580}(209,·)$, $\chi_{580}(129,·)$, $\chi_{580}(459,·)$, $\chi_{580}(1,·)$, $\chi_{580}(9,·)$, $\chi_{580}(339,·)$, $\chi_{580}(139,·)$, $\chi_{580}(141,·)$, $\chi_{580}(401,·)$, $\chi_{580}(531,·)$, $\chi_{580}(149,·)$, $\chi_{580}(151,·)$, $\chi_{580}(281,·)$, $\chi_{580}(411,·)$, $\chi_{580}(199,·)$, $\chi_{580}(231,·)$, $\chi_{580}(289,·)$, $\chi_{580}(219,·)$, $\chi_{580}(81,·)$, $\chi_{580}(161,·)$, $\chi_{580}(71,·)$, $\chi_{580}(109,·)$, $\chi_{580}(239,·)$, $\chi_{580}(91,·)$, $\chi_{580}(51,·)$, $\chi_{580}(181,·)$, $\chi_{580}(59,·)$, $\chi_{580}(469,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{17} a^{15} + \frac{7}{17} a^{13} - \frac{2}{17} a^{11} + \frac{3}{17} a^{9} + \frac{4}{17} a^{7} - \frac{6}{17} a^{5} - \frac{8}{17} a^{3} - \frac{5}{17} a$, $\frac{1}{17} a^{16} + \frac{7}{17} a^{14} - \frac{2}{17} a^{12} + \frac{3}{17} a^{10} + \frac{4}{17} a^{8} - \frac{6}{17} a^{6} - \frac{8}{17} a^{4} - \frac{5}{17} a^{2}$, $\frac{1}{17} a^{17} + \frac{1}{17} a$, $\frac{1}{17} a^{18} + \frac{1}{17} a^{2}$, $\frac{1}{17} a^{19} + \frac{1}{17} a^{3}$, $\frac{1}{17} a^{20} + \frac{1}{17} a^{4}$, $\frac{1}{17} a^{21} + \frac{1}{17} a^{5}$, $\frac{1}{17} a^{22} + \frac{1}{17} a^{6}$, $\frac{1}{17} a^{23} + \frac{1}{17} a^{7}$, $\frac{1}{17051} a^{24} - \frac{375}{17051} a^{22} + \frac{178}{17051} a^{20} + \frac{279}{17051} a^{18} + \frac{405}{17051} a^{16} + \frac{4705}{17051} a^{14} + \frac{618}{17051} a^{12} + \frac{2524}{17051} a^{10} - \frac{79}{17051} a^{8} - \frac{401}{1003} a^{6} + \frac{6730}{17051} a^{4} + \frac{2385}{17051} a^{2} + \frac{4}{59}$, $\frac{1}{17051} a^{25} - \frac{375}{17051} a^{23} + \frac{178}{17051} a^{21} + \frac{279}{17051} a^{19} + \frac{405}{17051} a^{17} - \frac{310}{17051} a^{15} - \frac{385}{17051} a^{13} - \frac{4497}{17051} a^{11} + \frac{1927}{17051} a^{9} + \frac{25}{59} a^{7} + \frac{2718}{17051} a^{5} + \frac{8403}{17051} a^{3} - \frac{463}{1003} a$, $\frac{1}{340532571829552933694635979} a^{26} - \frac{527456634554128931294}{340532571829552933694635979} a^{24} + \frac{391356588901876737172757}{20031327754679584334978587} a^{22} + \frac{4782418411027257570682945}{340532571829552933694635979} a^{20} - \frac{9049790686602605740496699}{340532571829552933694635979} a^{18} + \frac{1459479274961338814325692}{340532571829552933694635979} a^{16} - \frac{162490238234833123125302075}{340532571829552933694635979} a^{14} - \frac{157503082471827184805599373}{340532571829552933694635979} a^{12} - \frac{111361964142743013585441404}{340532571829552933694635979} a^{10} - \frac{46761847940908268667235868}{340532571829552933694635979} a^{8} + \frac{90633762127330264655712826}{340532571829552933694635979} a^{6} - \frac{98527981733187923792743972}{340532571829552933694635979} a^{4} - \frac{87559976294264684943780810}{340532571829552933694635979} a^{2} - \frac{266738955045537072512434}{1178313397334093196175211}$, $\frac{1}{53463613777239810590057848703} a^{27} + \frac{1517299970439871005124910}{53463613777239810590057848703} a^{25} - \frac{1243597366800683388128411589}{53463613777239810590057848703} a^{23} - \frac{1487801141981528490045428399}{53463613777239810590057848703} a^{21} - \frac{727361620549574300432595242}{53463613777239810590057848703} a^{19} - \frac{1026389288643622397361155822}{53463613777239810590057848703} a^{17} + \frac{628956907916908898420925666}{53463613777239810590057848703} a^{15} - \frac{24599177997718865804376735030}{53463613777239810590057848703} a^{13} - \frac{16612163209206672075230848313}{53463613777239810590057848703} a^{11} + \frac{26414901715780020558258908965}{53463613777239810590057848703} a^{9} + \frac{2483528643737189762840951490}{53463613777239810590057848703} a^{7} - \frac{432403952917817900687989348}{3144918457484694740591638159} a^{5} + \frac{1769701594866435838302150074}{53463613777239810590057848703} a^{3} - \frac{93387380960170412093448899}{3144918457484694740591638159} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{2436}$, which has order $311808$ (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:  \( 73869644668.60387 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{14}$ (as 28T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 28
The 28 conjugacy class representatives for $C_2\times C_{14}$
Character table for $C_2\times C_{14}$ is not computed

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-5}, \sqrt{-29})\), 7.7.594823321.1, 14.14.801611618199890796015625.1, 14.0.168110140833113738264576.1, 14.0.452882922503000372480000000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{4}$ R ${\href{/LocalNumberField/7.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/13.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/23.14.0.1}{14} }^{2}$ R ${\href{/LocalNumberField/31.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{4}$ ${\href{/LocalNumberField/53.14.0.1}{14} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}$

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$2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 2 x^{13} + x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{9} + 4 x^{8} - 2 x^{6} + 4 x^{5} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 1$$2$$7$$14$$C_{14}$$[2]^{7}$
$5$5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
5.14.7.1$x^{14} - 250 x^{8} + 15625 x^{2} - 312500$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$29$29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} - 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$