Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-82x-92\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-82xz^2-92z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-105651x-3963762\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(14, 32\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([14:32:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(507, 8532\right) \) | $0$ | $3$ |
Integral points
\( \left(14, 32\right) \), \( \left(14, -47\right) \)
\([14:32:1]\), \([14:-47:1]\)
\((507,\pm 8532)\)
Invariants
| Conductor: | $N$ | = | \( 158 \) | = | $2 \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $31554496$ | = | $2^{6} \cdot 79^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{59914169497}{31554496} \) | = | $2^{-6} \cdot 7^{3} \cdot 13^{3} \cdot 43^{3} \cdot 79^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.13055192504773046877559769391$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.13055192504773046877559769391$ |
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| $abc$ quality: | $Q$ | ≈ | $0.967983438848399$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.901869239932755$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $1.6852937385345253773046129288$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.1235291590230169182030752858 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.123529159 \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 1.685294 \cdot 1.000000 \cdot 6}{3^2} \\ & \approx 1.123529159\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 40 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $79$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3Cs.1.1 | 3.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2844 = 2^{2} \cdot 3^{2} \cdot 79 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1777 & 18 \\ 2529 & 511 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2827 & 18 \\ 2826 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 1423 & 18 \\ 1431 & 163 \end{array}\right),\left(\begin{array}{rr} 1433 & 18 \\ 2682 & 2737 \end{array}\right)$.
The torsion field $K:=\Q(E[2844])$ is a degree-$99664680960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2844\Z)$.
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 | $4$ | \( 79 \) |
| $3$ | good | $2$ | \( 1 \) |
| $79$ | split multiplicative | $80$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 158d
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-24964.2-d3 |
| $3$ | 3.3.316.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.2696112.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.4784690387019843.4 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.618118076690282564775869645105523.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.7592702318398778686518177792.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.1845690687107956693899710346370769989632.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 79 |
|---|---|---|---|
| Reduction type | nonsplit | ord | split |
| $\lambda$-invariant(s) | 2 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.