Normalized defining polynomial
\( x^{30} - 4x - 2 \)
Invariants
Degree: | $30$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 14]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(2960340693597750333450202133711692605134571100381102676639744\) \(\medspace = 2^{59}\cdot 31\cdot 101\cdot 36470152085223887711\cdot 44972943052468912543\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(103.68\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(31\), \(101\), \(36470152085223887711\), \(44972943052468912543\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{10270\!\cdots\!15126}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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);
oscar: torsion_units_generator(OK)
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Fundamental units: | $2a^{29}-2a^{28}+2a^{27}-2a^{25}+a^{24}+2a^{23}-3a^{22}+3a^{20}-2a^{19}-a^{18}+2a^{17}-a^{16}-a^{15}+2a^{14}-3a^{12}+2a^{11}+3a^{10}-5a^{9}+6a^{7}-4a^{6}-3a^{5}+5a^{4}-a^{3}-2a^{2}+3a-9$, $2a^{29}+2a^{28}-5a^{27}+2a^{26}+4a^{25}-6a^{24}+2a^{23}+3a^{22}-3a^{21}-2a^{20}+4a^{19}+a^{18}-7a^{17}+6a^{16}+a^{15}-7a^{14}+6a^{13}+a^{12}-5a^{11}+7a^{9}-5a^{8}-6a^{7}+13a^{6}-7a^{5}-7a^{4}+12a^{3}-2a^{2}-10a-3$, $4a^{29}-a^{28}-4a^{27}+4a^{26}-a^{25}-2a^{24}+5a^{23}-3a^{22}-4a^{21}+6a^{20}-3a^{19}-a^{18}+7a^{17}-7a^{16}-3a^{15}+8a^{14}-6a^{13}+2a^{12}+8a^{11}-12a^{10}+8a^{8}-9a^{7}+7a^{6}+7a^{5}-16a^{4}+4a^{3}+6a^{2}-10a-5$, $5a^{29}-5a^{28}+5a^{27}-a^{26}-3a^{25}+9a^{24}-15a^{23}+20a^{22}-23a^{21}+27a^{20}-27a^{19}+28a^{18}-25a^{17}+20a^{16}-12a^{15}+5a^{14}+5a^{13}-9a^{12}+16a^{11}-19a^{10}+22a^{9}-22a^{8}+20a^{7}-12a^{6}+7a^{5}+4a^{4}-7a^{3}+12a^{2}-13a-5$, $2a^{29}-a^{28}-2a^{23}+2a^{22}-a^{21}+2a^{20}-a^{19}+a^{17}-a^{16}+a^{15}-5a^{14}+5a^{13}-4a^{12}+5a^{11}-5a^{10}+6a^{9}-3a^{8}+2a^{7}-3a^{6}-2a^{3}+2a^{2}-a-3$, $3a^{29}+3a^{28}-4a^{26}-3a^{25}+2a^{24}+4a^{23}-a^{22}-3a^{21}+a^{20}+3a^{19}-5a^{17}-4a^{16}+3a^{15}+7a^{14}+4a^{13}-6a^{12}-11a^{11}+11a^{9}+6a^{8}-7a^{7}-9a^{6}+4a^{5}+9a^{4}-2a^{3}-11a^{2}-8a-5$, $3a^{29}+5a^{28}-a^{27}+4a^{26}+4a^{25}-4a^{24}+a^{23}-a^{22}-9a^{21}-2a^{20}-3a^{19}-7a^{18}+3a^{17}+a^{16}-a^{15}+11a^{14}+6a^{13}+2a^{12}+12a^{11}+a^{10}-7a^{9}+2a^{8}-10a^{7}-15a^{6}-3a^{5}-14a^{4}-11a^{3}+8a^{2}-a-3$, $118a^{29}-54a^{28}+24a^{27}-7a^{26}-a^{25}+3a^{24}-9a^{23}+7a^{22}-14a^{21}+8a^{20}-13a^{19}+9a^{18}-11a^{17}+13a^{16}-8a^{15}+12a^{14}-6a^{13}+10a^{12}-9a^{11}+5a^{10}-7a^{9}-4a^{8}-8a^{7}-6a^{6}-7a^{5}-9a^{4}+5a^{3}-7a^{2}+13a-475$, $a^{29}+4a^{28}+5a^{27}+10a^{26}-5a^{25}-4a^{24}-12a^{23}-5a^{22}+9a^{21}+6a^{20}+14a^{19}-3a^{18}-5a^{17}-12a^{16}-17a^{15}+7a^{14}+12a^{13}+23a^{12}+4a^{11}-13a^{10}-11a^{9}-24a^{8}-2a^{7}+12a^{6}+30a^{5}+27a^{4}-17a^{3}-22a^{2}-35a-15$, $8a^{29}-5a^{28}-2a^{25}+3a^{24}-a^{23}+4a^{22}-a^{21}+a^{20}-a^{19}-3a^{18}-3a^{16}+3a^{15}-2a^{14}+6a^{13}-a^{12}+3a^{11}-a^{10}-3a^{9}-2a^{8}-6a^{7}+3a^{6}-3a^{5}+8a^{4}+2a^{3}+6a^{2}-a-33$, $3a^{29}+2a^{27}-2a^{26}+a^{25}+4a^{24}-a^{22}+a^{21}+2a^{20}+6a^{19}-6a^{18}+a^{17}+8a^{16}-a^{15}+a^{13}+2a^{12}+8a^{11}-3a^{10}-a^{9}+6a^{8}+4a^{7}+6a^{6}-8a^{5}+5a^{4}+15a^{3}-2a^{2}-6a-1$, $a^{27}-2a^{26}-3a^{25}-3a^{24}-4a^{23}-3a^{22}-3a^{21}-4a^{20}-3a^{19}+2a^{17}-a^{16}+a^{15}+2a^{14}-2a^{13}+a^{12}-3a^{11}-12a^{10}-6a^{9}-7a^{8}-11a^{7}-6a^{6}-9a^{5}-6a^{4}+3a^{3}+a^{2}+2a+1$, $7a^{29}-8a^{28}+8a^{27}-7a^{26}+4a^{25}-4a^{23}+8a^{22}-9a^{21}+9a^{20}-8a^{19}+7a^{18}-9a^{17}+11a^{16}-14a^{15}+18a^{14}-20a^{13}+22a^{12}-22a^{11}+21a^{10}-23a^{9}+23a^{8}-23a^{7}+22a^{6}-19a^{5}+17a^{4}-13a^{3}+11a^{2}-12a-13$, $a^{29}+a^{28}-a^{27}+a^{25}-a^{24}+a^{23}+a^{22}-a^{21}+a^{20}-2a^{18}+a^{17}+a^{16}-2a^{15}+a^{14}-a^{13}-3a^{12}+a^{11}-a^{10}-a^{9}+4a^{8}-3a^{6}+a^{5}-3a^{4}-2a^{3}+4a^{2}-1$, $13a^{29}+43a^{28}-32a^{26}-34a^{25}-25a^{24}+34a^{23}+37a^{22}+25a^{21}-9a^{20}-47a^{19}-14a^{18}-7a^{17}+27a^{16}+18a^{15}-7a^{14}+12a^{13}-7a^{12}+17a^{11}-20a^{10}-48a^{9}-20a^{8}+6a^{7}+99a^{6}+68a^{5}-10a^{4}-106a^{3}-153a^{2}-5a+59$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 9880263253170928000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 9880263253170928000 \cdot 1}{2\cdot\sqrt{2960340693597750333450202133711692605134571100381102676639744}}\cr\approx \mathstrut & 1.71650795849983 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 265252859812191058636308480000000 |
The 5604 conjugacy class representatives for $S_{30}$ are not computed |
Character table for $S_{30}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.7.0.1}{7} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $29{,}\,{\href{/padicField/11.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{3}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $28{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ | $26{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $30$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $30$ | $30$ | $1$ | $59$ | |||
\(31\) | 31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Deg $28$ | $1$ | $28$ | $0$ | $C_{28}$ | $[\ ]^{28}$ | ||
\(101\) | $\Q_{101}$ | $x + 99$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
101.2.1.2 | $x^{2} + 202$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
101.3.0.1 | $x^{3} + 3 x + 99$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
101.5.0.1 | $x^{5} + 2 x + 99$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
101.9.0.1 | $x^{9} + 64 x^{2} + 47 x + 99$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
101.10.0.1 | $x^{10} + x^{6} + 67 x^{5} + 49 x^{4} + 100 x^{3} + 100 x^{2} + 52 x + 2$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | |
\(36470152085223887711\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(44972943052468912543\) | $\Q_{44972943052468912543}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{44972943052468912543}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ |