Properties

Label 30.0.131...536.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.310\times 10^{39}$
Root discriminant $20.13$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{10}\times S_3$ (as 30T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 6*x^28 + 25*x^26 + 45*x^24 - 44*x^22 - 197*x^20 - 20*x^18 + 411*x^16 + 283*x^14 - 424*x^12 - 397*x^10 + 348*x^8 + 358*x^6 + 8*x^4 - 7*x^2 + 1)
 
gp: K = bnfinit(x^30 + 6*x^28 + 25*x^26 + 45*x^24 - 44*x^22 - 197*x^20 - 20*x^18 + 411*x^16 + 283*x^14 - 424*x^12 - 397*x^10 + 348*x^8 + 358*x^6 + 8*x^4 - 7*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -7, 0, 8, 0, 358, 0, 348, 0, -397, 0, -424, 0, 283, 0, 411, 0, -20, 0, -197, 0, -44, 0, 45, 0, 25, 0, 6, 0, 1]);
 

\( x^{30} + 6 x^{28} + 25 x^{26} + 45 x^{24} - 44 x^{22} - 197 x^{20} - 20 x^{18} + 411 x^{16} + 283 x^{14} - 424 x^{12} - 397 x^{10} + 348 x^{8} + 358 x^{6} + 8 x^{4} - 7 x^{2} + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1310417368511817590988945406032175169536\)\(\medspace = -\,2^{40}\cdot 11^{26}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.13$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{806438113501246246} a^{28} - \frac{8547019296480505}{403219056750623123} a^{26} - \frac{161059070594040737}{806438113501246246} a^{24} - \frac{87862912953318957}{403219056750623123} a^{22} + \frac{147397953066571973}{806438113501246246} a^{20} - \frac{4901431061596493}{403219056750623123} a^{18} - \frac{87429740741882642}{403219056750623123} a^{16} - \frac{160823540610087811}{403219056750623123} a^{14} + \frac{113679147513814544}{403219056750623123} a^{12} - \frac{180267560084387693}{403219056750623123} a^{10} + \frac{182171665704510766}{403219056750623123} a^{8} + \frac{97550603202531440}{403219056750623123} a^{6} - \frac{151433037274049413}{806438113501246246} a^{4} + \frac{54798993027099655}{403219056750623123} a^{2} - \frac{201580180073151327}{403219056750623123}$, $\frac{1}{806438113501246246} a^{29} - \frac{8547019296480505}{403219056750623123} a^{27} - \frac{161059070594040737}{806438113501246246} a^{25} - \frac{87862912953318957}{403219056750623123} a^{23} + \frac{147397953066571973}{806438113501246246} a^{21} - \frac{4901431061596493}{403219056750623123} a^{19} - \frac{87429740741882642}{403219056750623123} a^{17} + \frac{81571975530447501}{806438113501246246} a^{15} - \frac{175860761722994035}{806438113501246246} a^{13} - \frac{1}{2} a^{12} - \frac{180267560084387693}{403219056750623123} a^{11} - \frac{1}{2} a^{10} + \frac{182171665704510766}{403219056750623123} a^{9} - \frac{1}{2} a^{8} - \frac{208117850345560243}{806438113501246246} a^{7} + \frac{125893009738286855}{403219056750623123} a^{5} + \frac{54798993027099655}{403219056750623123} a^{3} + \frac{58696604320469}{806438113501246246} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{90395440773}{291211635746} a^{29} - \frac{277418055635}{145605817873} a^{27} - \frac{1164367872014}{145605817873} a^{25} - \frac{2176299246142}{145605817873} a^{23} + \frac{1760559259543}{145605817873} a^{21} + \frac{9223935112516}{145605817873} a^{19} + \frac{3783920726191}{291211635746} a^{17} - \frac{18737732908828}{145605817873} a^{15} - \frac{14874663307821}{145605817873} a^{13} + \frac{18141256073509}{145605817873} a^{11} + \frac{20163995858949}{145605817873} a^{9} - \frac{14393368446386}{145605817873} a^{7} - \frac{35782616547667}{291211635746} a^{5} - \frac{1743953091134}{145605817873} a^{3} - \frac{7625965931}{291211635746} a \) (order $4$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 18931497.119988322 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 18931497.119988322 \cdot 1}{4\sqrt{1310417368511817590988945406032175169536}}\approx 0.122777326819508$ (assuming GRH)

Galois group

$C_{10}\times S_3$ (as 30T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 30 conjugacy class representatives for $C_{10}\times S_3$
Character table for $C_{10}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.44.1, \(\Q(\zeta_{11})^+\), 6.0.30976.1, 10.0.219503494144.1, 15.5.35351257235385344.1

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

Sibling fields

Degree 30 sibling: 30.10.14414591053629993500878399466353926864896.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $30$ $15^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{3}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
1.44.2t1.a.a$1$ $ 2^{2} \cdot 11 $ \(\Q(\sqrt{11}) \) $C_2$ (as 2T1) $1$ $1$
1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
1.44.10t1.b.a$1$ $ 2^{2} \cdot 11 $ \(\Q(\zeta_{44})^+\) $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.11.10t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.44.10t1.a.a$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
1.11.10t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.44.10t1.b.b$1$ $ 2^{2} \cdot 11 $ \(\Q(\zeta_{44})^+\) $C_{10}$ (as 10T1) $0$ $1$
* 1.44.10t1.a.b$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.44.10t1.b.c$1$ $ 2^{2} \cdot 11 $ \(\Q(\zeta_{44})^+\) $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.44.10t1.b.d$1$ $ 2^{2} \cdot 11 $ \(\Q(\zeta_{44})^+\) $C_{10}$ (as 10T1) $0$ $1$
1.11.10t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.44.10t1.a.c$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
1.11.10t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.44.10t1.a.d$1$ $ 2^{2} \cdot 11 $ 10.0.219503494144.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 2.176.6t3.a.a$2$ $ 2^{4} \cdot 11 $ 6.2.340736.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.44.3t2.b.a$2$ $ 2^{2} \cdot 11 $ 3.1.44.1 $S_3$ (as 3T2) $1$ $0$
* 2.484.15t4.a.a$2$ $ 2^{2} \cdot 11^{2}$ 15.5.35351257235385344.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.a.b$2$ $ 2^{2} \cdot 11^{2}$ 15.5.35351257235385344.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.a.c$2$ $ 2^{2} \cdot 11^{2}$ 15.5.35351257235385344.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.1936.30t12.a.a$2$ $ 2^{4} \cdot 11^{2}$ 30.0.1310417368511817590988945406032175169536.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.1936.30t12.a.b$2$ $ 2^{4} \cdot 11^{2}$ 30.0.1310417368511817590988945406032175169536.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.1936.30t12.a.c$2$ $ 2^{4} \cdot 11^{2}$ 30.0.1310417368511817590988945406032175169536.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.1936.30t12.a.d$2$ $ 2^{4} \cdot 11^{2}$ 30.0.1310417368511817590988945406032175169536.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.484.15t4.a.d$2$ $ 2^{2} \cdot 11^{2}$ 15.5.35351257235385344.1 $S_3 \times C_5$ (as 15T4) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.