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Properties

Label 15.3.22713113169...9441.1
Degree $15$
Signature $[3, 6]$
Discriminant $7^{12}\cdot 71^{12}$
Root discriminant $143.58$
Ramified primes $7, 71$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86089913, -46619874, -76618654, -16258641, 4806214, -1313053, -1939966, -344645, 216818, -17885, -20398, 1855, 482, -81, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 3*x^14 - 81*x^13 + 482*x^12 + 1855*x^11 - 20398*x^10 - 17885*x^9 + 216818*x^8 - 344645*x^7 - 1939966*x^6 - 1313053*x^5 + 4806214*x^4 - 16258641*x^3 - 76618654*x^2 - 46619874*x + 86089913)
 
gp: K = bnfinit(x^15 - 3*x^14 - 81*x^13 + 482*x^12 + 1855*x^11 - 20398*x^10 - 17885*x^9 + 216818*x^8 - 344645*x^7 - 1939966*x^6 - 1313053*x^5 + 4806214*x^4 - 16258641*x^3 - 76618654*x^2 - 46619874*x + 86089913, 1)
 

Normalized defining polynomial

\( x^{15} - 3 x^{14} - 81 x^{13} + 482 x^{12} + 1855 x^{11} - 20398 x^{10} - 17885 x^{9} + 216818 x^{8} - 344645 x^{7} - 1939966 x^{6} - 1313053 x^{5} + 4806214 x^{4} - 16258641 x^{3} - 76618654 x^{2} - 46619874 x + 86089913 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(227131131690502261690628246149441=7^{12}\cdot 71^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $143.58$
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:  $7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{71} a^{5} - \frac{1}{71} a^{4} - \frac{28}{71} a^{3} + \frac{34}{71} a^{2} + \frac{25}{71} a - \frac{1}{71}$, $\frac{1}{142} a^{6} - \frac{1}{142} a^{5} + \frac{43}{142} a^{4} - \frac{37}{142} a^{3} + \frac{25}{142} a^{2} - \frac{1}{142} a - \frac{1}{2}$, $\frac{1}{142} a^{7} + \frac{24}{71} a^{4} + \frac{14}{71} a^{3} + \frac{8}{71} a^{2} + \frac{7}{71} a - \frac{29}{142}$, $\frac{1}{142} a^{8} - \frac{33}{71} a^{4} - \frac{30}{71} a^{3} - \frac{28}{71} a^{2} + \frac{49}{142} a + \frac{24}{71}$, $\frac{1}{994} a^{9} - \frac{3}{994} a^{8} + \frac{3}{994} a^{7} - \frac{1}{994} a^{6} - \frac{1}{142} a^{5} - \frac{35}{142} a^{4} - \frac{55}{142} a^{3} + \frac{16}{71} a^{2} - \frac{12}{71} a + \frac{25}{71}$, $\frac{1}{70574} a^{10} - \frac{1}{35287} a^{9} - \frac{9}{5041} a^{8} - \frac{89}{70574} a^{7} - \frac{15}{70574} a^{6} + \frac{35}{10082} a^{5} - \frac{65}{10082} a^{4} + \frac{308}{5041} a^{3} - \frac{3653}{10082} a^{2} + \frac{4973}{10082} a - \frac{2064}{5041}$, $\frac{1}{70574} a^{11} + \frac{6}{35287} a^{9} + \frac{227}{70574} a^{8} + \frac{233}{70574} a^{7} + \frac{73}{70574} a^{6} + \frac{5}{10082} a^{5} - \frac{1958}{5041} a^{4} - \frac{2563}{10082} a^{3} - \frac{913}{10082} a^{2} + \frac{1418}{5041} a - \frac{2282}{5041}$, $\frac{1}{4940180} a^{12} - \frac{1}{494018} a^{11} + \frac{17}{4940180} a^{10} - \frac{697}{2470090} a^{9} + \frac{897}{705740} a^{8} + \frac{517}{352870} a^{7} + \frac{1807}{705740} a^{6} - \frac{1143}{176435} a^{5} + \frac{329857}{705740} a^{4} - \frac{7233}{176435} a^{3} + \frac{8835}{20164} a^{2} + \frac{20983}{50410} a - \frac{26989}{100820}$, $\frac{1}{4940180} a^{13} - \frac{13}{4940180} a^{11} - \frac{17}{2470090} a^{10} + \frac{739}{4940180} a^{9} + \frac{376}{176435} a^{8} - \frac{1363}{705740} a^{7} + \frac{159}{352870} a^{6} - \frac{269}{100820} a^{5} - \frac{40808}{176435} a^{4} + \frac{15307}{141148} a^{3} - \frac{11401}{25205} a^{2} - \frac{44929}{100820} a + \frac{1498}{5041}$, $\frac{1}{371678139972735578715460} a^{14} - \frac{3024816662500147}{371678139972735578715460} a^{13} - \frac{2816466717436763}{185839069986367789357730} a^{12} - \frac{1451229565029624603}{371678139972735578715460} a^{11} + \frac{544951355403663913}{185839069986367789357730} a^{10} - \frac{166379106849453739363}{371678139972735578715460} a^{9} + \frac{78837623618422456959}{26548438569481112765390} a^{8} - \frac{64394030338172751033}{53096877138962225530780} a^{7} - \frac{112604866717200929}{74784333998538345818} a^{6} - \frac{3571196858449822635}{1517053632541777872308} a^{5} + \frac{344129760415336206799}{26548438569481112765390} a^{4} + \frac{23329709158761968100023}{53096877138962225530780} a^{3} + \frac{1013556326124287504427}{3792634081354444680770} a^{2} - \frac{553967634151431749683}{1517053632541777872308} a + \frac{3078797230702429643}{13569352706098192060}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  $8$
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:  \( 1666688074.98 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_6$ (as 15T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 10 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 15 sibling: data not computed
Degree 25 sibling: data not computed
Degree 30 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$71$71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$
71.5.4.4$x^{5} + 568$$5$$1$$4$$C_5$$[\ ]_{5}$