# $p$-adic fields downloaded from the LMFDB on 11 March 2026. # Search link: https://www.lmfdb.org/padicField/?p=17&n=12 # Query "{'p': 17, 'n': 12}" returned 17 fields, sorted by prime. # Each entry in the following data list has the form: # [Label, Polynomial, $p$, $f$, $e$, $c$, Galois group, Artin slope content] # For more details, see the definitions at the bottom of the file. "17.12.1.0a1.1" [3, 9, 14, 6, 13, 14, 14, 4, 1, 0, 0, 0, 1] 17 12 1 0 [12, 1] [[], 1, 12] "17.6.2.6a1.1" [9, 35, 69, 60, 112, 12, 46, 6, 24, 0, 4, 0, 1] 17 6 2 6 [12, 1] [[], 2, 6] "17.6.2.6a1.2" [26, 18, 69, 60, 112, 12, 46, 6, 24, 0, 4, 0, 1] 17 6 2 6 [12, 2] [[], 2, 6] "17.4.3.8a1.1" [27, 287, 1089, 2260, 2568, 1650, 769, 420, 156, 30, 21, 0, 1] 17 4 3 8 [12, 19] [[], 3, 12] "17.4.3.8a1.2" [44, 270, 1089, 2260, 2568, 1650, 769, 420, 156, 30, 21, 0, 1] 17 4 3 8 [12, 5] [[], 3, 4] "17.3.4.9a1.1" [38416, 10976, 1193, 11032, 2353, 168, 1180, 168, 6, 56, 4, 0, 1] 17 3 4 9 [12, 1] [[], 4, 3] "17.3.4.9a1.2" [38416, 10993, 1176, 11032, 2353, 168, 1180, 168, 6, 56, 4, 0, 1] 17 3 4 9 [12, 1] [[], 4, 3] "17.3.4.9a1.3" [38433, 10976, 1176, 11032, 2353, 168, 1180, 168, 6, 56, 4, 0, 1] 17 3 4 9 [12, 1] [[], 4, 3] "17.3.4.9a1.4" [38467, 11248, 1176, 11032, 2353, 168, 1180, 168, 6, 56, 4, 0, 1] 17 3 4 9 [12, 1] [[], 4, 3] "17.2.6.10a1.1" [729, 23345, 312498, 2250720, 9263295, 21112128, 22883356, 7037376, 1029255, 83360, 3858, 96, 1] 17 2 6 10 [12, 19] [[], 6, 6] "17.2.6.10a1.2" [746, 23328, 312498, 2250720, 9263295, 21112128, 22883356, 7037376, 1029255, 83360, 3858, 96, 1] 17 2 6 10 [12, 3] [[], 6, 2] "17.2.6.10a1.3" [967, 23345, 312498, 2250720, 9263295, 21112128, 22883356, 7037376, 1029255, 83360, 3858, 96, 1] 17 2 6 10 [12, 18] [[], 6, 6] "17.2.6.10a1.4" [967, 23583, 312498, 2250720, 9263295, 21112128, 22883356, 7037376, 1029255, 83360, 3858, 96, 1] 17 2 6 10 [12, 5] [[], 6, 2] "17.1.12.11a1.1" [17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 17 1 12 11 [12, 11] [[], 12, 2] "17.1.12.11a1.2" [51, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 17 1 12 11 [12, 11] [[], 12, 2] "17.1.12.11a1.3" [153, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 17 1 12 11 [12, 11] [[], 12, 2] "17.1.12.11a1.4" [170, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 17 1 12 11 [12, 11] [[], 12, 2] # Label -- # The label associated to a $p$-adic field # $K$ has the form $p.f.e.c\ell m.n$, where # - $p$ is the characteristic of the # residue field of $K$. # - $f$ is the residue field degree # of $K/\Q_p$. # - $e$ is the ramification index of # $K/\Q_p$, # - $c$ is the discriminant exponent # of $K/\Q_p$, # - $\ell$ is a string of one or more letters used to identify the # absolute family $I/\Q_p$ that contains # $K/\Q_p$, # - $m$ identifies the subfamily of $I/\Q_p$ that # contains $K/\Q_p$, # - $n$ is a number used to distinguish fields in the same subfamily. # Thus if $K$ has label $p.f.e.c\ell m.n$ then $K/\Q_p$ is in the absolute # family with label $p.f.e.c\ell$. #Polynomial (coeffs) -- # The **defining polynomial** of a $p$-adic field $K$ is an irreducible polynomial $f(x)\in\mathbb{Q}_p[x]$ such that $K\cong \mathbb{Q}_p(a)$, where $a$ is a root of $f(x)$. # The defining polynomial can be chosen to be monic with coefficients in $\mathbb{Z}_p$; by Krasner's lemma, we can further take $f(x)\in \mathbb{Z}[x]$. # The LMFDB uses the following conventions for choosing defining polynomials: # 1. For unramified extensions, the polynomial needs to be irreducible modulo $p$. When it is feasible to compute, we use the Conway polynomial for the residue prime $p$ and the given degree. # 2. For totally ramified extensions, we pick an Eisenstein polynomial which is reduced in the sense of Monge. # 3. In the remaining cases, we pick a polynomial which is in Eisenstein form. #$p$ (p) -- # If $p$ is a prime number, then every rational number $r$ can be written in the form $r=p^j \frac{a}{b}$ where $\gcd(p, ab)=1$. We define $|r|_p = p^{-j}$, and define a metric on $\Q$ by $d(u,v) = |u-v|_p$; in other words, $u$ and $v$ are close to each other if they are congruent modulo a larger power of $p$. # The completion of $\Q$ with respect to this metric is $\Q_p$, which is itself a field in the natural way. Any non-zero element $x \in \Q_p$ can be written uniquely as the sum of a convergent series $x = \sum_{n=n_0}^{+\infty} a_n p^n$, where $a_n \in \{ 0, \cdots, p-1 \}$, $n_0 \in \Z$ is such that $|x|_p=p^{-n_0}$, and $a_{n_0} \neq 0$. # Example: the element $a = 3 + 7 + 2 \cdot 7^2 + 6 \cdot 7^3 + 7^4 + 2 \cdot 7^5 + \cdots$ satisfies $a^2=2$, since truncating the expansion after $n$ terms and squaring yields a number which is congruent to $2 \bmod 7^n$. #$f$ (f) -- # Let $L/K$ be an extension of $p$-adic fields. # Let $\kappa$ be the residue field of $K$ # and let $\lambda$ be the residue field of $L$. The **residue field degree** $f$ # of $L/K$ is the degree of the field extension $\lambda/\kappa$. # The **base residue field degree** $f_0$ of $L/K$ is the residue field degree of $K/\Q_p$. The **absolute residue field degree** $f_{\mathrm{abs}}$ of $L/K$ is the # residue field degree of $L$ over $\Q_p$. #$e$ (e) -- # Let $L/K$ be a finite extension of $p$-adic fields. # Then $\mathcal{O}_L$ and $\mathcal{O}_K$, the rings of integers of $L$ and $K$, # are discrete valuation domains, so they have unique maximal ideals $P_L$ and # $P_K$ which are principal. If $P_K=(\pi_K)$, the element $\pi_K$ is a # **uniformizer** for $K$. # We have $\pi_K\mathcal{O}_L=P_L^e$ for some positive integer $e$, which factors as $e = p^w e_{\mathrm{tame}}$ with $e_{\mathrm{tame}}$ coprime to $p$. The integer $e$ # is the **ramification index**, $w$ is the **wild ramification exponent** and $e_{\mathrm{tame}}$ is the **tame ramification index** of $L/K$. If $e=1$ we say that the # extension $L/K$ is **unramified**, and if $e=[L:K]$ we say that $L/K$ # is **totally ramified**. # The **base ramification index** $e_0$ of $L/K$ is the ramification index of $K/\Q_p$. # The **absolute ramification index** $e_{\mathrm{abs}}$ of $L/K$ is the ramification index of # $L$ over $\Q_p$. #$c$ (c) -- # Let $L/K$ be a finite extension of $p$-adic fields, with rings of integers $\mathcal{O}_L$ and $\mathcal{O}_K$ and uniformizers $\pi_L$ and $\pi_K$. The **discriminant** of $L/K$ is the square of the determinant of the matrix # \[ # \left( \begin{array}{ccc} # \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\ # \vdots & & \vdots \\ # \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\ # \end{array} \right) # \] # where $\sigma_1,..., \sigma_n$ are the embeddings of $L$ into an algebraic closure $\overline{K}$, and $\{\beta_1, \ldots, \beta_n\}$ is a basis for $\mathcal{O}_L$ as a free $\mathcal{O}_K$-module. # The discriminant of $L/K$ is an element of $K^\times$ which is well-defined up to the square of a unit. Thus, it is of the form $\pi_K^c u$ where $u \in \mathcal{O}_K^\times$ is a unit. The value $c$ is the **discriminant exponent** for $L/K$. Together with the discriminant root field of $L/K$, it determines the discriminant of $L/K$ (up to the square of a unit). # The **base discriminant exponent** $c_0$ of $L/K$ is the discriminant exponent of $K/\Q_p$. The **absolute discriminant exponent** $c_{\mathrm{abs}}$ of $L/K$ is the discriminant exponent of $L/\Q_p$. #Galois group (gal) -- # Let $K$ be a finite degree $n$ separable extension of a field $F$, and $K^{gal}$ be its # Galois (or normal) closure. # The **Galois group** for $K/F$ is the automorphism group $\Aut(K^{gal}/F)$. # This automorphism group acts on the $n$ embeddings $K\hookrightarrow K^{gal}$ via composition. As a result, we get an injection $\Aut(K^{gal}/F)\hookrightarrow S_n$, which is well-defined up to the labelling of the $n$ embeddings, which corresponds to being well-defined up to conjugation in $S_n$. # We use the notation $\Gal(K/F)$ for $\Aut(K/F)$ when $K=K^{gal}$. # There is a naming convention for Galois groups up to degree $47$. #Artin slope content (slopes) -- # Let $L/K$ be an extension of $p$-adic fields. # For each subextension $E/K$ of $L/K$ we plot the point \((n_E,c_E)\), # where $n_E=[E:K]$ and $c_E$ is the # discriminant exponent of $E/K$. # Let $B$ be the boundary of the lower convex hull of these points. The slopes # of the segments of $B$ are called the **Artin slopes** of # $L/K$, and those which are greater than 1 are the **wild Artin slopes**. If a segment corresponding to a wild slope runs from \((n_1, c_1)\) to \((n_2, c_2)\) then $n_2/n_1=p^m$ # for some $m\in\N$, and the corresponding Artin slope is repeated $m$ # times. The **Swan slopes** of $L/K$ are obtained by subtracting $1$ from each Artin slope, and the **wild Swan slopes** are precisely the positive ones. The Artin slopes and Swan # slopes of $L/K$ are also referred to as the **visible Artin slopes** and the # **visible Swan slopes**. This is to distinguish them from the # hidden slopes of $L/K$. # Let $f$ be the residue field degree, let $\epsilon$ be the tame degree of $L/K$, and let $a_1\le a_2\le\dots\le a_w$ be the wild Artin slopes of $L/K$. # The **Artin slope content** of $L/K$ is the collection of data # $[a_1,\dots,a_w]_{\epsilon}^f$. The **Swan slope content** of $L/K$ is $[s_1,\dots,s_w]_{\epsilon}^f$, where $s_1,\dots,s_w$ are the wild Swan slopes of # $L/K$. # The Swan slopes are the same as the positive upper ramification jumps of # $L/K$. In the case where $L/K$ is a Galois extension these are defined for # instance in Chapter IV of Serre's Local Fields # \cite{doi:10.1007/978-1-4757-5673-9,MR:0554237}. For the general case where # $L/K$ is separable but not necessarily Galois see the appendix to Deligne # \cite{MR:0771673}.