Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1077x+13877\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1077xz^2+13877z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-17235x+870894\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(19, -8)$ | $0$ | $3$ |
Integral points
\( \left(19, -8\right) \), \( \left(19, -11\right) \)
Invariants
| Conductor: | $N$ | = | \( 162 \) | = | $2 \cdot 3^{4}$ |
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| Discriminant: | $\Delta$ | = | $-93312$ | = | $-1 \cdot 2^{7} \cdot 3^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{189613868625}{128} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.26936807887764018183084025072$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.27993806545641466386678236774$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1259568215438134$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.39986514729382$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $2.8006777425532496677500992009$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.93355924751774988925003306697 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.933559248 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.800678 \cdot 1.000000 \cdot 3}{3^2} \\ & \approx 0.933559248\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 42 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $3$ | $3$ | $IV$ | additive | -1 | 4 | 6 | 0 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2G | 8.2.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 465 & 442 \\ 448 & 201 \end{array}\right),\left(\begin{array}{rr} 281 & 392 \\ 112 & 393 \end{array}\right),\left(\begin{array}{rr} 73 & 402 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 426 \\ 42 & 253 \end{array}\right),\left(\begin{array}{rr} 1 & 216 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 462 & 1 \end{array}\right),\left(\begin{array}{rr} 85 & 42 \\ 189 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 168 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 168 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 295 & 42 \\ 231 & 211 \end{array}\right),\left(\begin{array}{rr} 22 & 321 \\ 189 & 169 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 168 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 463 & 282 \\ 420 & 295 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$15676416$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 81 = 3^{4} \) |
| $3$ | additive | $6$ | \( 2 \) |
| $7$ | good | $2$ | \( 81 = 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 162c
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.648.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.6.330812181.2 | \(\Z/21\Z\) | not in database |
| $9$ | 9.3.167365651248.1 | \(\Z/9\Z\) | not in database |
| $12$ | 12.2.51998697814228992.41 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.41451359947637504606208.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.9490397425838961457555240648704.3 | \(\Z/42\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | ss | ord |
| $\lambda$-invariant(s) | 4 | - | 0,0 | 0 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.