Normalized defining polynomial
\( x^{30} - 5 x^{29} + 8 x^{28} + 19 x^{27} - 63 x^{26} - 52 x^{25} + 265 x^{24} - 211 x^{23} - 393 x^{22} + 858 x^{21} - 86 x^{20} - 2380 x^{19} + 4186 x^{18} + 8272 x^{17} - 3336 x^{16} - 13036 x^{15} + 7176 x^{14} + 23259 x^{13} + 6583 x^{12} - 9115 x^{11} - 7328 x^{10} + 1279 x^{9} + 4440 x^{8} + 2429 x^{7} - 349 x^{6} - 1189 x^{5} - 88 x^{4} + 271 x^{3} + 100 x^{2} - 13 x - 1 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(301337616246914973122131519082482771026611328125\)\(\medspace = 5^{15}\cdot 439^{14}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $38.25$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $5, 439$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $2$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{4}{27} a^{12} + \frac{1}{27} a^{11} - \frac{4}{27} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{5}{27} a^{7} - \frac{4}{27} a^{6} + \frac{5}{27} a^{5} + \frac{1}{3} a^{4} - \frac{13}{27} a^{3} - \frac{8}{27} a - \frac{8}{27}$, $\frac{1}{27} a^{21} + \frac{1}{27} a^{19} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} - \frac{4}{27} a^{13} + \frac{1}{27} a^{12} - \frac{4}{27} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{27} a^{8} - \frac{4}{27} a^{7} + \frac{5}{27} a^{6} + \frac{1}{3} a^{5} - \frac{13}{27} a^{4} - \frac{8}{27} a^{2} - \frac{8}{27} a + \frac{1}{3}$, $\frac{1}{27} a^{22} - \frac{1}{27} a^{19} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{4}{27} a^{14} - \frac{4}{27} a^{11} + \frac{4}{27} a^{10} - \frac{4}{27} a^{9} - \frac{4}{27} a^{8} + \frac{4}{27} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{5}{27} a^{3} + \frac{13}{27} a^{2} - \frac{4}{27} a - \frac{13}{27}$, $\frac{1}{27} a^{23} + \frac{1}{27} a^{19} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} + \frac{4}{27} a^{14} + \frac{4}{27} a^{13} + \frac{4}{27} a^{12} - \frac{1}{27} a^{11} + \frac{4}{27} a^{10} - \frac{1}{27} a^{9} + \frac{2}{9} a^{7} - \frac{7}{27} a^{6} + \frac{11}{27} a^{5} - \frac{7}{27} a^{4} - \frac{1}{9} a^{3} + \frac{2}{27} a^{2} - \frac{4}{9} a + \frac{7}{27}$, $\frac{1}{27} a^{24} - \frac{1}{9} a^{8} + \frac{2}{27}$, $\frac{1}{27} a^{25} - \frac{1}{9} a^{9} + \frac{2}{27} a$, $\frac{1}{81} a^{26} + \frac{1}{81} a^{25} - \frac{1}{81} a^{22} - \frac{2}{81} a^{19} - \frac{1}{81} a^{18} + \frac{4}{81} a^{17} - \frac{2}{81} a^{16} - \frac{1}{27} a^{15} - \frac{7}{81} a^{14} + \frac{2}{27} a^{13} + \frac{2}{27} a^{12} - \frac{11}{81} a^{11} - \frac{10}{81} a^{10} + \frac{1}{81} a^{9} - \frac{11}{81} a^{8} + \frac{13}{27} a^{7} + \frac{8}{81} a^{6} - \frac{11}{27} a^{5} - \frac{5}{27} a^{4} - \frac{32}{81} a^{3} + \frac{1}{81} a^{2} + \frac{4}{27} a + \frac{13}{81}$, $\frac{1}{729} a^{27} + \frac{2}{729} a^{26} + \frac{10}{729} a^{25} - \frac{4}{243} a^{24} + \frac{11}{729} a^{23} - \frac{7}{729} a^{22} + \frac{1}{81} a^{21} + \frac{7}{729} a^{20} + \frac{11}{243} a^{19} - \frac{4}{243} a^{18} + \frac{29}{729} a^{17} - \frac{38}{729} a^{16} - \frac{13}{729} a^{15} + \frac{86}{729} a^{14} - \frac{10}{243} a^{13} - \frac{101}{729} a^{12} + \frac{2}{81} a^{11} - \frac{40}{243} a^{10} - \frac{115}{729} a^{9} + \frac{43}{729} a^{8} + \frac{326}{729} a^{7} + \frac{182}{729} a^{6} - \frac{80}{243} a^{5} - \frac{131}{729} a^{4} - \frac{88}{729} a^{3} - \frac{167}{729} a^{2} + \frac{220}{729} a - \frac{47}{729}$, $\frac{1}{384183} a^{28} - \frac{260}{384183} a^{27} - \frac{847}{384183} a^{26} - \frac{2965}{384183} a^{25} - \frac{112}{384183} a^{24} + \frac{1}{279} a^{23} - \frac{4088}{384183} a^{22} + \frac{1132}{384183} a^{21} + \frac{5597}{384183} a^{20} + \frac{448}{14229} a^{19} + \frac{18815}{384183} a^{18} - \frac{14665}{384183} a^{17} - \frac{20954}{384183} a^{16} + \frac{334}{42687} a^{15} + \frac{53290}{384183} a^{14} + \frac{60436}{384183} a^{13} - \frac{29842}{384183} a^{12} + \frac{18671}{128061} a^{11} + \frac{16835}{384183} a^{10} + \frac{58487}{384183} a^{9} - \frac{35114}{384183} a^{8} + \frac{12025}{42687} a^{7} + \frac{85303}{384183} a^{6} + \frac{11051}{22599} a^{5} - \frac{166322}{384183} a^{4} + \frac{7787}{384183} a^{3} + \frac{10474}{42687} a^{2} - \frac{52430}{128061} a - \frac{56923}{384183}$, $\frac{1}{6932787919812818813867113376921047991049} a^{29} - \frac{2153464308234134186111767009577}{2760966913505702434833577609287553959} a^{28} + \frac{970866685316710843053001252395220706}{6932787919812818813867113376921047991049} a^{27} - \frac{33093227870933222437200936937984443946}{6932787919812818813867113376921047991049} a^{26} - \frac{63260015356179361793332499850241201937}{6932787919812818813867113376921047991049} a^{25} + \frac{112786505585563284222044559681333556342}{6932787919812818813867113376921047991049} a^{24} - \frac{40306421851876579696840883628992433827}{2310929306604272937955704458973682663683} a^{23} + \frac{117398551423359694093899420270077188538}{6932787919812818813867113376921047991049} a^{22} - \frac{50740576993151739325591307068645201112}{6932787919812818813867113376921047991049} a^{21} - \frac{7101360271869078289308228435849127367}{407811054106636400815712551583591058297} a^{20} + \frac{298686993682945136523790333949380241543}{6932787919812818813867113376921047991049} a^{19} + \frac{41670478074274955750636118241830360332}{2310929306604272937955704458973682663683} a^{18} + \frac{160629448965266180846727285419116589542}{6932787919812818813867113376921047991049} a^{17} - \frac{128139266846633417657081684006436781891}{2310929306604272937955704458973682663683} a^{16} + \frac{13204749581417629795335256186484005111}{770309768868090979318568152991227554561} a^{15} - \frac{292584915461408707535096495633993477974}{6932787919812818813867113376921047991049} a^{14} - \frac{1126925107580585852878934794282464997514}{6932787919812818813867113376921047991049} a^{13} + \frac{59366859038605883876112624759289008911}{6932787919812818813867113376921047991049} a^{12} + \frac{195349447251961042896658894096099465856}{6932787919812818813867113376921047991049} a^{11} + \frac{14575708550595851788280901405743425432}{135937018035545466938570850527863686099} a^{10} + \frac{198696386812724344793597817304947872731}{6932787919812818813867113376921047991049} a^{9} + \frac{31383265940718125495287212902493520715}{770309768868090979318568152991227554561} a^{8} + \frac{322087799195464009567506905642194300139}{2310929306604272937955704458973682663683} a^{7} - \frac{1860244480352380588947016078422033983488}{6932787919812818813867113376921047991049} a^{6} + \frac{304042728037087395132654496874621507314}{770309768868090979318568152991227554561} a^{5} + \frac{1547545954021488284910428209144879051637}{6932787919812818813867113376921047991049} a^{4} - \frac{425515906131768765781378707840133084640}{2310929306604272937955704458973682663683} a^{3} + \frac{1778951606093810719888709702795166272038}{6932787919812818813867113376921047991049} a^{2} - \frac{96591288373017859824994113018363798506}{256769922956030326439522717663742518187} a + \frac{1624677260981336965511385422630217276962}{6932787919812818813867113376921047991049}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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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)];
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Regulator: | \( 748323688575.9082 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
A solvable group of order 60 |
The 18 conjugacy class representatives for $D_{30}$ |
Character table for $D_{30}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 3.1.439.1, 5.1.192721.1, 6.2.24090125.1, 10.2.116066824503125.1, 15.1.3142328914862177479.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | $30$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{15}$ | R | $30$ | $15^{2}$ | ${\href{/LocalNumberField/13.6.0.1}{6} }^{5}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{15}$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ | $15^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ | $30$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
$5$ | 5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
5.10.5.1 | $x^{10} - 50 x^{6} + 625 x^{2} - 12500$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
439 | Data not computed |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.2195.2t1.a.a | $1$ | $ 5 \cdot 439 $ | \(\Q(\sqrt{-2195}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.439.2t1.a.a | $1$ | $ 439 $ | \(\Q(\sqrt{-439}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 2.10975.6t3.a.a | $2$ | $ 5^{2} \cdot 439 $ | 6.0.10575564875.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
* | 2.439.3t2.a.a | $2$ | $ 439 $ | 3.1.439.1 | $S_3$ (as 3T2) | $1$ | $0$ |
* | 2.439.5t2.a.a | $2$ | $ 439 $ | 5.1.192721.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.439.5t2.a.b | $2$ | $ 439 $ | 5.1.192721.1 | $D_{5}$ (as 5T2) | $1$ | $0$ |
* | 2.10975.10t3.a.b | $2$ | $ 5^{2} \cdot 439 $ | 10.0.50953335956871875.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.10975.10t3.a.a | $2$ | $ 5^{2} \cdot 439 $ | 10.0.50953335956871875.1 | $D_{10}$ (as 10T3) | $1$ | $0$ |
* | 2.439.15t2.a.a | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.10975.30t14.a.c | $2$ | $ 5^{2} \cdot 439 $ | 30.2.301337616246914973122131519082482771026611328125.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
* | 2.439.15t2.a.c | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.10975.30t14.a.a | $2$ | $ 5^{2} \cdot 439 $ | 30.2.301337616246914973122131519082482771026611328125.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
* | 2.439.15t2.a.b | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.10975.30t14.a.b | $2$ | $ 5^{2} \cdot 439 $ | 30.2.301337616246914973122131519082482771026611328125.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |
* | 2.439.15t2.a.d | $2$ | $ 439 $ | 15.1.3142328914862177479.1 | $D_{15}$ (as 15T2) | $1$ | $0$ |
* | 2.10975.30t14.a.d | $2$ | $ 5^{2} \cdot 439 $ | 30.2.301337616246914973122131519082482771026611328125.1 | $D_{30}$ (as 30T14) | $1$ | $0$ |