Properties

Label 15.5.77595502265817783.1
Degree $15$
Signature $[5, 5]$
Discriminant $-\,3\cdot 13^{3}\cdot 11772948303113$
Root discriminant $13.37$
Ramified primes $3, 13, 11772948303113$
Class number $1$
Class group Trivial
Galois Group 15T104

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -7, -3, 7, 20, -4, -10, -15, 3, 12, 1, -2, -3, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*x + 1)
gp: K = bnfinit(x^15 - 3*x^13 - 2*x^12 + x^11 + 12*x^10 + 3*x^9 - 15*x^8 - 10*x^7 - 4*x^6 + 20*x^5 + 7*x^4 - 3*x^3 - 7*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{15} \) \(\mathstrut -\mathstrut 3 x^{13} \) \(\mathstrut -\mathstrut 2 x^{12} \) \(\mathstrut +\mathstrut x^{11} \) \(\mathstrut +\mathstrut 12 x^{10} \) \(\mathstrut +\mathstrut 3 x^{9} \) \(\mathstrut -\mathstrut 15 x^{8} \) \(\mathstrut -\mathstrut 10 x^{7} \) \(\mathstrut -\mathstrut 4 x^{6} \) \(\mathstrut +\mathstrut 20 x^{5} \) \(\mathstrut +\mathstrut 7 x^{4} \) \(\mathstrut -\mathstrut 3 x^{3} \) \(\mathstrut -\mathstrut 7 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[5, 5]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-77595502265817783=-\,3\cdot 13^{3}\cdot 11772948303113\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $13.37$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 13, 11772948303113$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{136709} a^{14} + \frac{29762}{136709} a^{13} + \frac{39030}{136709} a^{12} - \frac{5515}{136709} a^{11} + \frac{50080}{136709} a^{10} - \frac{57255}{136709} a^{9} + \frac{54378}{136709} a^{8} + \frac{36879}{136709} a^{7} - \frac{43773}{136709} a^{6} + \frac{64740}{136709} a^{5} + \frac{15254}{136709} a^{4} - \frac{21034}{136709} a^{3} - \frac{23400}{136709} a^{2} - \frac{35161}{136709} a + \frac{45709}{136709}$

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

Class group and class number

Trivial Abelian group, order $1$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $9$
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:  \( \frac{23519}{136709} a^{14} + \frac{22398}{136709} a^{13} - \frac{54365}{136709} a^{12} - \frac{107153}{136709} a^{11} - \frac{53224}{136709} a^{10} + \frac{276723}{136709} a^{9} + \frac{276905}{136709} a^{8} - \frac{198113}{136709} a^{7} - \frac{488544}{136709} a^{6} - \frac{318200}{136709} a^{5} + \frac{444537}{136709} a^{4} + \frac{324643}{136709} a^{3} + \frac{45834}{136709} a^{2} - \frac{272236}{136709} a - \frac{49605}{136709} \),  \( \frac{29105}{136709} a^{14} + \frac{34786}{136709} a^{13} - \frac{83640}{136709} a^{12} - \frac{154418}{136709} a^{11} - \frac{12958}{136709} a^{10} + \frac{349353}{136709} a^{9} + \frac{401724}{136709} a^{8} - \frac{349191}{136709} a^{7} - \frac{705539}{136709} a^{6} - \frac{139156}{136709} a^{5} + \frac{346965}{136709} a^{4} + \frac{535168}{136709} a^{3} + \frac{27238}{136709} a^{2} - \frac{367458}{136709} a - \frac{91543}{136709} \),  \( \frac{7345}{136709} a^{14} + \frac{4199}{136709} a^{13} - \frac{3423}{136709} a^{12} - \frac{41811}{136709} a^{11} - \frac{46319}{136709} a^{10} + \frac{115618}{136709} a^{9} + \frac{79421}{136709} a^{8} + \frac{55726}{136709} a^{7} - \frac{246535}{136709} a^{6} - \frac{232020}{136709} a^{5} + \frac{349377}{136709} a^{4} - \frac{13560}{136709} a^{3} + \frac{243631}{136709} a^{2} - \frac{287662}{136709} a - \frac{24699}{136709} \),  \( \frac{40610}{136709} a^{14} - \frac{9449}{136709} a^{13} - \frac{132555}{136709} a^{12} - \frac{34808}{136709} a^{11} + \frac{65716}{136709} a^{10} + \frac{431249}{136709} a^{9} + \frac{30103}{136709} a^{8} - \frac{674450}{136709} a^{7} - \frac{131112}{136709} a^{6} - \frac{96088}{136709} a^{5} + \frac{583297}{136709} a^{4} + \frac{240510}{136709} a^{3} - \frac{146450}{136709} a^{2} - \frac{99414}{136709} a - \frac{129021}{136709} \),  \( \frac{3675}{136709} a^{14} + \frac{8150}{136709} a^{13} + \frac{27509}{136709} a^{12} - \frac{34693}{136709} a^{11} - \frac{103023}{136709} a^{10} - \frac{16974}{136709} a^{9} + \frac{107301}{136709} a^{8} + \frac{325124}{136709} a^{7} - \frac{95991}{136709} a^{6} - \frac{364287}{136709} a^{5} - \frac{128949}{136709} a^{4} - \frac{59365}{136709} a^{3} + \frac{268379}{136709} a^{2} - \frac{26670}{136709} a + \frac{101923}{136709} \),  \( \frac{41850}{136709} a^{14} - \frac{15999}{136709} a^{13} - \frac{130341}{136709} a^{12} - \frac{37958}{136709} a^{11} + \frac{99030}{136709} a^{10} + \frac{523729}{136709} a^{9} - \frac{75423}{136709} a^{8} - \frac{742005}{136709} a^{7} - \frac{272868}{136709} a^{6} + \frac{70038}{136709} a^{5} + \frac{1042542}{136709} a^{4} - \frac{3649}{136709} a^{3} - \frac{180142}{136709} a^{2} - \frac{499010}{136709} a - \frac{47387}{136709} \),  \( a \),  \( \frac{1475}{136709} a^{14} + \frac{15361}{136709} a^{13} + \frac{14761}{136709} a^{12} - \frac{68794}{136709} a^{11} - \frac{91569}{136709} a^{10} + \frac{35037}{136709} a^{9} + \frac{232785}{136709} a^{8} + \frac{259761}{136709} a^{7} - \frac{311945}{136709} a^{6} - \frac{478218}{136709} a^{5} - \frac{57335}{136709} a^{4} + \frac{281211}{136709} a^{3} + \frac{482504}{136709} a^{2} - \frac{49764}{136709} a - \frac{113471}{136709} \),  \( \frac{4199}{136709} a^{14} + \frac{18612}{136709} a^{13} - \frac{27121}{136709} a^{12} - \frac{53664}{136709} a^{11} + \frac{27478}{136709} a^{10} + \frac{57386}{136709} a^{9} + \frac{165901}{136709} a^{8} - \frac{173085}{136709} a^{7} - \frac{202640}{136709} a^{6} + \frac{202477}{136709} a^{5} - \frac{64975}{136709} a^{4} + \frac{128957}{136709} a^{3} - \frac{236247}{136709} a^{2} + \frac{4681}{136709} a + \frac{129364}{136709} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 416.964032767 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

15T104:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 1307674368000
The 176 conjugacy class representatives for S15 are not computed
Character table for S15 is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 30 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 $15$ R ${\href{/LocalNumberField/5.11.0.1}{11} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.13.0.1}{13} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ R ${\href{/LocalNumberField/17.11.0.1}{11} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.13.0.1}{13} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.13.0.1}{13} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.13.0.1}{13} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $15$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$

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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.8.0.1$x^{8} - x^{3} + 2$$1$$8$$0$$C_8$$[\ ]^{8}$
$13$13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.3.4$x^{4} + 104$$4$$1$$3$$C_4$$[\ ]_{4}$
13.7.0.1$x^{7} - 10 x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
11772948303113Data not computed