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Properties

Label 14.14.5435954051...0625.1
Degree $14$
Signature $[14, 0]$
Discriminant $3^{6}\cdot 5^{10}\cdot 140327041^{6}$
Root discriminant $15{,}681.11$
Ramified primes $3, 5, 140327041$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_7:D_7.C_2$ (as 14T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169982622681922057876, 18209668860129269820, -1753790683418315657, -231103082160131005, 3823892392283781, 998854565092922, 9352873618820, -1712039577669, -36184911805, 1149789082, 34468746, -213950, -10562, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 10562*x^12 - 213950*x^11 + 34468746*x^10 + 1149789082*x^9 - 36184911805*x^8 - 1712039577669*x^7 + 9352873618820*x^6 + 998854565092922*x^5 + 3823892392283781*x^4 - 231103082160131005*x^3 - 1753790683418315657*x^2 + 18209668860129269820*x + 169982622681922057876)
 
gp: K = bnfinit(x^14 - 10562*x^12 - 213950*x^11 + 34468746*x^10 + 1149789082*x^9 - 36184911805*x^8 - 1712039577669*x^7 + 9352873618820*x^6 + 998854565092922*x^5 + 3823892392283781*x^4 - 231103082160131005*x^3 - 1753790683418315657*x^2 + 18209668860129269820*x + 169982622681922057876, 1)
 

Normalized defining polynomial

\( x^{14} - 10562 x^{12} - 213950 x^{11} + 34468746 x^{10} + 1149789082 x^{9} - 36184911805 x^{8} - 1712039577669 x^{7} + 9352873618820 x^{6} + 998854565092922 x^{5} + 3823892392283781 x^{4} - 231103082160131005 x^{3} - 1753790683418315657 x^{2} + 18209668860129269820 x + 169982622681922057876 \)

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

Invariants

Degree:  $14$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54359540516805627060781106472576003494442683387594619140625=3^{6}\cdot 5^{10}\cdot 140327041^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15{,}681.11$
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:  $3, 5, 140327041$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{15} a^{6} + \frac{1}{15} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{3} a + \frac{7}{15}$, $\frac{1}{15} a^{7} - \frac{1}{15} a^{5} - \frac{4}{15} a^{2} + \frac{1}{3} a - \frac{1}{15}$, $\frac{1}{15} a^{8} + \frac{1}{15} a^{5} + \frac{2}{5} a^{4} + \frac{1}{3} a^{3} - \frac{4}{15} a^{2} + \frac{4}{15} a + \frac{7}{15}$, $\frac{1}{825} a^{9} - \frac{2}{165} a^{8} + \frac{1}{275} a^{7} - \frac{2}{165} a^{6} - \frac{23}{275} a^{5} + \frac{191}{825} a^{4} + \frac{14}{55} a^{3} - \frac{202}{825} a^{2} - \frac{59}{165} a + \frac{7}{25}$, $\frac{1}{14850} a^{10} - \frac{1}{14850} a^{9} - \frac{14}{825} a^{8} - \frac{184}{7425} a^{7} - \frac{163}{4950} a^{6} - \frac{91}{990} a^{5} + \frac{267}{550} a^{4} + \frac{2659}{7425} a^{3} - \frac{728}{2475} a^{2} - \frac{3359}{14850} a + \frac{292}{675}$, $\frac{1}{29700} a^{11} + \frac{17}{29700} a^{9} + \frac{32}{1485} a^{8} - \frac{47}{29700} a^{7} + \frac{1}{75} a^{6} - \frac{151}{4950} a^{5} - \frac{653}{2700} a^{4} - \frac{571}{2970} a^{3} + \frac{5053}{29700} a^{2} - \frac{71}{5940} a + \frac{427}{1350}$, $\frac{1}{148500} a^{12} - \frac{1}{74250} a^{11} + \frac{1}{148500} a^{10} + \frac{1}{14850} a^{9} - \frac{43}{5940} a^{8} - \frac{563}{24750} a^{7} + \frac{391}{24750} a^{6} - \frac{703}{148500} a^{5} - \frac{1972}{7425} a^{4} + \frac{589}{5940} a^{3} + \frac{529}{1500} a^{2} + \frac{5162}{37125} a - \frac{49}{375}$, $\frac{1}{13495238355602148564263057946501672759945072119544803401127318236943657257760003649000} a^{13} + \frac{33285796206460190881748613940111748203365717962983534409465230522305837470935001}{13495238355602148564263057946501672759945072119544803401127318236943657257760003649000} a^{12} - \frac{6937589850169754874676484485453855697582534665922420570612212514657413765428793}{2699047671120429712852611589300334551989014423908960680225463647388731451552000729800} a^{11} + \frac{452588707346990098879237775324027579685076968725901889572034487077543314807067213}{13495238355602148564263057946501672759945072119544803401127318236943657257760003649000} a^{10} + \frac{15439911704705211584669565051656637655440811715585950153566396328102839675192991}{2699047671120429712852611589300334551989014423908960680225463647388731451552000729800} a^{9} + \frac{301070050710844668563256820168152545788208440413952329908497158623099041427823999397}{13495238355602148564263057946501672759945072119544803401127318236943657257760003649000} a^{8} + \frac{1052793616086789667949590688530061117264264628613506938989646817269175183965405693}{306709962627321558278705862420492562726024366353290986389257232657810392221818264750} a^{7} + \frac{12497926811019391495843374574857201728736132656361086993959218174908675428108726607}{539809534224085942570522317860066910397802884781792136045092729477746290310400145960} a^{6} + \frac{46784026973507890524175612425142995922704842790647699639419636931763747418970051659}{1499470928400238729362561994055741417771674679949422600125257581882628584195555961000} a^{5} - \frac{22733279490399861905428695252553457391658395203501909805438704188666430379854874131}{299894185680047745872512398811148283554334935989884520025051516376525716839111192200} a^{4} - \frac{258677889929608463690262821130060612476206598364280722945506886936405312607571063659}{2249206392600358094043842991083612126657512019924133900187886372823942876293333941500} a^{3} + \frac{5332512120337755606795985985513261573073850907507263512443019960666802614595978343341}{13495238355602148564263057946501672759945072119544803401127318236943657257760003649000} a^{2} - \frac{23096360039272656397148547170848679110349884488568866966587319182218264101306773344}{337380958890053714106576448662541818998626802988620085028182955923591431444000091225} a - \frac{30945535020901422785037359076364284460830800148539513732522854911886201868054737603}{306709962627321558278705862420492562726024366353290986389257232657810392221818264750}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 7791759503400000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:D_7.C_2$ (as 14T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 196
The 16 conjugacy class representatives for $C_7:D_7.C_2$
Character table for $C_7:D_7.C_2$

Intermediate fields

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

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

Sibling fields

Degree 14 siblings: data not computed
Degree 28 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{7}$

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
$3$3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
140327041Data not computed