Normalized defining polynomial
\( x^{25} - 4x - 4 \)
Invariants
Degree: | $25$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1400610626524185559330987516514007343169536\) \(\medspace = 2^{24}\cdot 7\cdot 19\cdot 2579\cdot 1081796927611\cdot 224982529004839973\) | 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: | \(48.51\) | 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))
| |
Galois root discriminant: | $2^{24/25}7^{1/2}19^{1/2}2579^{1/2}1081796927611^{1/2}224982529004839973^{1/2}\approx 5.620663122436464e+17$ | ||
Ramified primes: | \(2\), \(7\), \(19\), \(2579\), \(1081796927611\), \(224982529004839973\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{83482\!\cdots\!90521}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$, $\frac{1}{2}a^{23}$, $\frac{1}{2}a^{24}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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)
| |
Fundamental units: | $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-3$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-1$, $\frac{1}{2}a^{17}-a-1$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{13}+a^{7}-2a-1$, $2a^{24}-2a^{23}+2a^{22}-2a^{21}+2a^{20}-2a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+a^{16}-a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-a^{5}-a^{3}-a-7$, $\frac{1}{2}a^{24}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}+a^{14}+a^{12}+a^{9}+a^{7}+a^{4}+2a^{2}+a-1$, $\frac{1}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+a^{13}+a^{11}-a^{10}-a^{8}+a^{7}+a^{6}+a^{5}-2a^{3}+1$, $\frac{1}{2}a^{24}-a^{23}+a^{22}-\frac{3}{2}a^{21}+\frac{1}{2}a^{20}-a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}+\frac{3}{2}a^{16}-\frac{1}{2}a^{15}+\frac{3}{2}a^{14}-\frac{3}{2}a^{13}+2a^{12}-a^{11}-2a^{9}+a^{8}-2a^{7}+2a^{4}-a^{3}+2a^{2}+2a-1$, $\frac{1}{2}a^{24}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+a^{18}-\frac{1}{2}a^{16}-a^{15}+\frac{1}{2}a^{14}+a^{13}-a^{11}-a^{10}+2a^{8}-2a^{6}-a^{5}+a^{4}+2a^{3}-3a-3$, $\frac{3}{2}a^{24}-\frac{3}{2}a^{23}+\frac{3}{2}a^{22}-\frac{3}{2}a^{21}+a^{20}-a^{19}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-a^{16}+2a^{15}-2a^{14}+\frac{3}{2}a^{13}-2a^{12}+a^{11}-a^{10}+a^{9}+a^{6}-2a^{5}+2a^{4}-3a^{3}+2a^{2}-a-5$, $a^{24}-a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+a^{18}+a^{16}+\frac{1}{2}a^{15}-a^{14}-a^{13}-a^{11}+2a^{9}+a^{8}+a^{7}-3a^{5}-3a^{4}+a^{3}+a^{2}+2a-1$, $\frac{1}{2}a^{24}-a^{23}+\frac{1}{2}a^{22}+a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{3}{2}a^{17}+\frac{1}{2}a^{16}+\frac{1}{2}a^{15}+a^{14}+\frac{1}{2}a^{13}-2a^{11}+a^{8}+a^{7}-a^{6}-2a^{5}-a^{4}+2a^{2}+a-3$ (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)]
| |
Regulator: | \( 208280915557.53195 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{12}\cdot 208280915557.53195 \cdot 1}{2\cdot\sqrt{1400610626524185559330987516514007343169536}}\cr\approx \mathstrut & 0.666268360041294 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 15511210043330985984000000 |
The 1958 conjugacy class representatives for $S_{25}$ are not computed |
Character table for $S_{25}$ is not computed |
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 | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.4.0.1}{4} }^{6}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | R | $16{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | $24{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.7.0.1}{7} }^{2}{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $21{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $20{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }{,}\,{\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 $25$ | $25$ | $1$ | $24$ | |||
\(7\) | 7.2.1.2 | $x^{2} + 7$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.23.0.1 | $x^{23} + 4 x^{2} + 4 x + 4$ | $1$ | $23$ | $0$ | $C_{23}$ | $[\ ]^{23}$ | |
\(19\) | $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
19.3.0.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
19.8.0.1 | $x^{8} + x^{4} + 12 x^{3} + 10 x^{2} + 3 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
19.9.0.1 | $x^{9} + 11 x^{3} + 14 x^{2} + 16 x + 17$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(2579\) | $\Q_{2579}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $22$ | $1$ | $22$ | $0$ | 22T1 | $[\ ]^{22}$ | ||
\(1081796927611\) | $\Q_{1081796927611}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(224982529004839973\) | $\Q_{224982529004839973}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ |