Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417)
gp: K = bnfinit(y^13 - 3*y^12 - 87*y^11 + 736*y^10 - 1956*y^9 - 15927*y^8 + 216019*y^7 - 1196967*y^6 + 4265724*y^5 - 10045872*y^4 + 13521123*y^3 - 7227009*y^2 + 1336176*y + 150417, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417)
\( x^{13} - 3 x^{12} - 87 x^{11} + 736 x^{10} - 1956 x^{9} - 15927 x^{8} + 216019 x^{7} - 1196967 x^{6} + \cdots + 150417 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[1, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(673089242845163403360476390625\)
\(\medspace = 3^{6}\cdot 5^{6}\cdot 79^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(197.00\) | 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: | $3^{1/2}5^{1/2}79^{12/13}\approx 218.62579719842265$ | ||
| Ramified primes: |
\(3\), \(5\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a^{2}$, $\frac{1}{9}a^{6}-\frac{4}{9}a^{3}+\frac{1}{3}a$, $\frac{1}{45}a^{7}-\frac{2}{45}a^{6}-\frac{2}{15}a^{5}+\frac{2}{45}a^{4}+\frac{14}{45}a^{3}-\frac{1}{15}a^{2}+\frac{7}{15}a+\frac{2}{5}$, $\frac{1}{135}a^{8}-\frac{1}{135}a^{7}-\frac{1}{45}a^{6}-\frac{4}{135}a^{5}-\frac{14}{135}a^{4}+\frac{17}{45}a^{3}+\frac{11}{45}a^{2}-\frac{4}{15}a-\frac{1}{5}$, $\frac{1}{1485}a^{9}+\frac{1}{297}a^{8}-\frac{2}{495}a^{7}+\frac{32}{1485}a^{6}-\frac{56}{1485}a^{5}-\frac{23}{165}a^{4}+\frac{7}{45}a^{3}+\frac{14}{55}a^{2}+\frac{1}{11}a-\frac{19}{55}$, $\frac{1}{51975}a^{10}-\frac{86}{51975}a^{8}-\frac{23}{3465}a^{7}+\frac{841}{17325}a^{6}-\frac{16}{2079}a^{5}-\frac{1114}{7425}a^{4}-\frac{502}{1155}a^{3}-\frac{2048}{17325}a^{2}-\frac{32}{105}a+\frac{502}{1925}$, $\frac{1}{155925}a^{11}+\frac{1}{155925}a^{10}+\frac{19}{155925}a^{9}+\frac{479}{155925}a^{8}-\frac{1147}{155925}a^{7}-\frac{2602}{155925}a^{6}+\frac{5189}{51975}a^{5}+\frac{2389}{51975}a^{4}+\frac{5149}{17325}a^{3}+\frac{461}{1925}a^{2}+\frac{697}{5775}a+\frac{144}{1925}$, $\frac{1}{33\!\cdots\!25}a^{12}+\frac{7345035024098}{33\!\cdots\!25}a^{11}+\frac{1090546606913}{47\!\cdots\!75}a^{10}+\frac{268610109160279}{11\!\cdots\!75}a^{9}+\frac{26\!\cdots\!26}{33\!\cdots\!25}a^{8}+\frac{31\!\cdots\!89}{33\!\cdots\!25}a^{7}-\frac{12\!\cdots\!52}{33\!\cdots\!25}a^{6}-\frac{60\!\cdots\!71}{36\!\cdots\!25}a^{5}-\frac{31725403619657}{220415376876975}a^{4}+\frac{54\!\cdots\!84}{12\!\cdots\!75}a^{3}+\frac{11\!\cdots\!23}{12\!\cdots\!75}a^{2}+\frac{26\!\cdots\!86}{12\!\cdots\!75}a+\frac{11\!\cdots\!28}{40\!\cdots\!25}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$, $5$ |
Class group and class number
$C_{13}$, which has order $13$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $6$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| 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: |
$\frac{91\!\cdots\!14}{11\!\cdots\!75}a^{12}-\frac{12\!\cdots\!68}{11\!\cdots\!75}a^{11}-\frac{39\!\cdots\!11}{52\!\cdots\!75}a^{10}+\frac{60\!\cdots\!62}{12\!\cdots\!75}a^{9}-\frac{81\!\cdots\!44}{11\!\cdots\!25}a^{8}-\frac{56\!\cdots\!33}{36\!\cdots\!25}a^{7}+\frac{17\!\cdots\!82}{11\!\cdots\!75}a^{6}-\frac{79\!\cdots\!76}{11\!\cdots\!75}a^{5}+\frac{57\!\cdots\!91}{269396571738525}a^{4}-\frac{15\!\cdots\!19}{40\!\cdots\!25}a^{3}+\frac{16\!\cdots\!96}{12\!\cdots\!75}a^{2}+\frac{52\!\cdots\!22}{12\!\cdots\!75}a+\frac{12\!\cdots\!66}{40\!\cdots\!25}$, $\frac{17\!\cdots\!36}{33\!\cdots\!25}a^{12}-\frac{12\!\cdots\!04}{11\!\cdots\!75}a^{11}-\frac{26\!\cdots\!17}{58\!\cdots\!75}a^{10}+\frac{11\!\cdots\!97}{33\!\cdots\!25}a^{9}-\frac{82\!\cdots\!58}{11\!\cdots\!75}a^{8}-\frac{97\!\cdots\!47}{11\!\cdots\!75}a^{7}+\frac{34\!\cdots\!83}{33\!\cdots\!25}a^{6}-\frac{22\!\cdots\!89}{40\!\cdots\!25}a^{5}+\frac{43\!\cdots\!31}{24\!\cdots\!25}a^{4}-\frac{12\!\cdots\!58}{33\!\cdots\!75}a^{3}+\frac{48\!\cdots\!83}{12\!\cdots\!75}a^{2}-\frac{42\!\cdots\!84}{12\!\cdots\!75}a-\frac{20\!\cdots\!02}{40\!\cdots\!25}$, $\frac{28\!\cdots\!21}{13\!\cdots\!85}a^{12}+\frac{37\!\cdots\!21}{66\!\cdots\!25}a^{11}-\frac{14\!\cdots\!44}{94\!\cdots\!75}a^{10}+\frac{45\!\cdots\!64}{66\!\cdots\!25}a^{9}-\frac{15\!\cdots\!02}{66\!\cdots\!25}a^{8}-\frac{23\!\cdots\!27}{66\!\cdots\!25}a^{7}+\frac{56\!\cdots\!77}{22\!\cdots\!75}a^{6}-\frac{23\!\cdots\!76}{22\!\cdots\!75}a^{5}+\frac{70\!\cdots\!23}{24\!\cdots\!25}a^{4}-\frac{11\!\cdots\!92}{24\!\cdots\!75}a^{3}+\frac{18\!\cdots\!33}{73\!\cdots\!25}a^{2}-\frac{40\!\cdots\!41}{81\!\cdots\!25}a-\frac{47\!\cdots\!17}{81\!\cdots\!25}$, $\frac{28\!\cdots\!23}{60\!\cdots\!75}a^{12}-\frac{96\!\cdots\!23}{66\!\cdots\!25}a^{11}-\frac{55\!\cdots\!37}{13\!\cdots\!25}a^{10}+\frac{21\!\cdots\!08}{60\!\cdots\!75}a^{9}-\frac{63\!\cdots\!01}{66\!\cdots\!25}a^{8}-\frac{45\!\cdots\!99}{60\!\cdots\!75}a^{7}+\frac{22\!\cdots\!37}{22\!\cdots\!75}a^{6}-\frac{12\!\cdots\!32}{22\!\cdots\!75}a^{5}+\frac{50\!\cdots\!11}{24\!\cdots\!25}a^{4}-\frac{36\!\cdots\!22}{73\!\cdots\!25}a^{3}+\frac{49\!\cdots\!73}{73\!\cdots\!25}a^{2}-\frac{31\!\cdots\!97}{81\!\cdots\!25}a+\frac{13\!\cdots\!98}{148576291080035}$, $\frac{14\!\cdots\!39}{25\!\cdots\!25}a^{12}-\frac{46\!\cdots\!83}{25\!\cdots\!25}a^{11}-\frac{18\!\cdots\!68}{36\!\cdots\!75}a^{10}+\frac{99\!\cdots\!43}{23\!\cdots\!75}a^{9}-\frac{27\!\cdots\!61}{25\!\cdots\!25}a^{8}-\frac{23\!\cdots\!44}{25\!\cdots\!25}a^{7}+\frac{10\!\cdots\!74}{84\!\cdots\!75}a^{6}-\frac{63\!\cdots\!03}{94\!\cdots\!75}a^{5}+\frac{57\!\cdots\!26}{24\!\cdots\!25}a^{4}-\frac{52\!\cdots\!64}{94\!\cdots\!75}a^{3}+\frac{21\!\cdots\!66}{28\!\cdots\!25}a^{2}-\frac{41\!\cdots\!81}{94\!\cdots\!75}a+\frac{32\!\cdots\!42}{31\!\cdots\!25}$, $\frac{29\!\cdots\!11}{33\!\cdots\!25}a^{12}-\frac{35\!\cdots\!57}{33\!\cdots\!25}a^{11}-\frac{37\!\cdots\!32}{47\!\cdots\!75}a^{10}+\frac{16\!\cdots\!17}{33\!\cdots\!25}a^{9}-\frac{22\!\cdots\!14}{33\!\cdots\!25}a^{8}-\frac{53\!\cdots\!76}{33\!\cdots\!25}a^{7}+\frac{18\!\cdots\!51}{11\!\cdots\!75}a^{6}-\frac{81\!\cdots\!83}{11\!\cdots\!75}a^{5}+\frac{57\!\cdots\!27}{269396571738525}a^{4}-\frac{12\!\cdots\!18}{36\!\cdots\!25}a^{3}+\frac{61\!\cdots\!28}{12\!\cdots\!75}a^{2}+\frac{66\!\cdots\!26}{12\!\cdots\!75}a+\frac{16\!\cdots\!58}{40\!\cdots\!25}$
| 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: | \( 22619543081.392696 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{6}\cdot 22619543081.392696 \cdot 13}{2\cdot\sqrt{673089242845163403360476390625}}\cr\approx \mathstrut & 22.0531212569358 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 3*x^12 - 87*x^11 + 736*x^10 - 1956*x^9 - 15927*x^8 + 216019*x^7 - 1196967*x^6 + 4265724*x^5 - 10045872*x^4 + 13521123*x^3 - 7227009*x^2 + 1336176*x + 150417);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A solvable group of order 26 |
| The 8 conjugacy class representatives for $D_{13}$ |
| Character table for $D_{13}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Galois closure: | deg 26 |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.13.0.1}{13} }$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.13.0.1}{13} }$ | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.13.0.1}{13} }$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(79\)
| 79.13.12.6 | $x^{13} + 1738$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.93615.13t2.a.f | $2$ | $ 3 \cdot 5 \cdot 79^{2}$ | 13.1.673089242845163403360476390625.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.93615.13t2.a.b | $2$ | $ 3 \cdot 5 \cdot 79^{2}$ | 13.1.673089242845163403360476390625.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.93615.13t2.a.e | $2$ | $ 3 \cdot 5 \cdot 79^{2}$ | 13.1.673089242845163403360476390625.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.93615.13t2.a.c | $2$ | $ 3 \cdot 5 \cdot 79^{2}$ | 13.1.673089242845163403360476390625.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.93615.13t2.a.a | $2$ | $ 3 \cdot 5 \cdot 79^{2}$ | 13.1.673089242845163403360476390625.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
| * | 2.93615.13t2.a.d | $2$ | $ 3 \cdot 5 \cdot 79^{2}$ | 13.1.673089242845163403360476390625.1 | $D_{13}$ (as 13T2) | $1$ | $0$ |
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 *.