Normalized defining polynomial
\( x^{26} - 5 \)
Invariants
Degree: | $26$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[2, 12]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(27338662801166956620931510927975177764892578125\)\(\medspace = 5^{25}\cdot 13^{26}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $61.10$ | 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: | $5, 13$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{8}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{9}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{10}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{11}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{12}$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $13$ | 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: | \( 9612138497230.408 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_2\times F_{13}$ (as 26T10):
A solvable group of order 312 |
The 26 conjugacy class representatives for $C_2\times F_{13}$ |
Character table for $C_2\times F_{13}$ is not computed |
Intermediate fields
\(\Q(\sqrt{5}) \), 13.1.73944117820374267578125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 26 sibling: | 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$ | ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ | R | ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ | ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ | ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ | ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | $26$ | ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 | Data not computed | ||||||
13 | Data not computed |