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Properties

Label 14.0.13760207184...6523.4
Degree $14$
Signature $[0, 7]$
Discriminant $-\,7^{25}\cdot 29^{13}$
Root discriminant $736.28$
Ramified primes $7, 29$
Class number $1383004$ (GRH)
Class group $[1383004]$ (GRH)
Galois group $C_{14}$ (as 14T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![525016637623, -488778538759, 88403705502, -10631790456, 8530487112, 1371017746, 410450166, 45617783, 6725390, 276080, 15834, -2233, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 2233*x^11 + 15834*x^10 + 276080*x^9 + 6725390*x^8 + 45617783*x^7 + 410450166*x^6 + 1371017746*x^5 + 8530487112*x^4 - 10631790456*x^3 + 88403705502*x^2 - 488778538759*x + 525016637623)
 
gp: K = bnfinit(x^14 - 2233*x^11 + 15834*x^10 + 276080*x^9 + 6725390*x^8 + 45617783*x^7 + 410450166*x^6 + 1371017746*x^5 + 8530487112*x^4 - 10631790456*x^3 + 88403705502*x^2 - 488778538759*x + 525016637623, 1)
 

Normalized defining polynomial

\( x^{14} - 2233 x^{11} + 15834 x^{10} + 276080 x^{9} + 6725390 x^{8} + 45617783 x^{7} + 410450166 x^{6} + 1371017746 x^{5} + 8530487112 x^{4} - 10631790456 x^{3} + 88403705502 x^{2} - 488778538759 x + 525016637623 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-13760207184971837367389621368916758066523=-\,7^{25}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $736.28$
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:  $7, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1421=7^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1421}(1184,·)$, $\chi_{1421}(1,·)$, $\chi_{1421}(36,·)$, $\chi_{1421}(6,·)$, $\chi_{1421}(1415,·)$, $\chi_{1421}(1385,·)$, $\chi_{1421}(1420,·)$, $\chi_{1421}(237,·)$, $\chi_{1421}(750,·)$, $\chi_{1421}(1296,·)$, $\chi_{1421}(1205,·)$, $\chi_{1421}(216,·)$, $\chi_{1421}(125,·)$, $\chi_{1421}(671,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{13} + \frac{96399567621966156268153919147983297282273253862251853987473265020335270}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{12} + \frac{2329726409576080457069352871429816853711385889100387771828089649977721}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{11} - \frac{37562037506772949768135496794468548516167839588460628250021744811854997}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{10} + \frac{18484092635762140301792886909407685741061650199627915075401947110451490}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{9} - \frac{66093372257527656274376458579261956871935983432727131965537648312462328}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{8} - \frac{28739826566534978051201396116849055490526911461172637780664640772945947}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{7} + \frac{96928644310213850295448568331619780505473243594461767246479741937539236}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{6} - \frac{18798672295696293726857066988425795673845774169541326245280677324863377}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{5} + \frac{92658714510049960249588048391484964453591265924134552109312907388667542}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{4} - \frac{40382752687399612311210959718612505430670320770469980153111771417505227}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{3} + \frac{79207351399838029148874532938629729683753251718757409062335148170756529}{201376763354354085589007560817035440584226870964866888681790878886620901} a^{2} - \frac{7239459799850928645796503966763407714337753141280360402927156204867771}{201376763354354085589007560817035440584226870964866888681790878886620901} a - \frac{242210223044674018620302771574665876614988568970663016842109541527824}{502186442280184752092288181588617058813533344052037128882271518420501}$

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

Class group and class number

$C_{1383004}$, which has order $1383004$ (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:  $6$
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:  \( 59811086.44175506 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{14}$ (as 14T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 14
The 14 conjugacy class representatives for $C_{14}$
Character table for $C_{14}$

Intermediate fields

\(\Q(\sqrt{-203}) \), 7.7.8233120419813614521.6

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 ${\href{/LocalNumberField/2.14.0.1}{14} }$ ${\href{/LocalNumberField/3.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }$ R ${\href{/LocalNumberField/11.14.0.1}{14} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{2}$ R ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/37.14.0.1}{14} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/43.14.0.1}{14} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }$

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
$7$7.14.25.52$x^{14} - 56 x^{13} - 112 x^{12} + 49 x^{11} + 98 x^{10} - 147 x^{9} - 42 x^{7} + 98 x^{6} - 49 x^{5} + 98 x^{4} - 98 x^{3} + 147 x - 84$$14$$1$$25$$C_{14}$$[2]_{2}$
$29$29.14.13.7$x^{14} - 118784$$14$$1$$13$$C_{14}$$[\ ]_{14}$