Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-3x-6\)
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(homogenize, simplify) |
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\(y^2z=x^3-3xz^2-6z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3x-6\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 864 \) | = | $2^{5} \cdot 3^{3}$ |
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| Discriminant: | $\Delta$ | = | $-13824$ | = | $-1 \cdot 2^{9} \cdot 3^{3} $ |
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| j-invariant: | $j$ | = | \( -216 \) | = | $-1 \cdot 2^{3} \cdot 3^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.52077557042309926344138257465$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3152890280100856683531179750$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2262943855309167$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.5299830469901075$ | |||
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: | $\Omega$ | ≈ | $1.6436559999231596206953307579$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6436559999231596206953307579 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.643656000 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.643656 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.643656000\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 48 |
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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 not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | 1 | 5 | 9 | 0 |
| $3$ | $1$ | $II$ | additive | -1 | 3 | 3 | 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 |
|---|---|---|
| $3$ | 3Nn | 9.9.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 72 = 2^{3} \cdot 3^{2} \), index $36$, genus $3$, and generators
$\left(\begin{array}{rr} 17 & 68 \\ 26 & 31 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 18 \\ 36 & 29 \end{array}\right),\left(\begin{array}{rr} 19 & 44 \\ 19 & 55 \end{array}\right),\left(\begin{array}{rr} 37 & 18 \\ 45 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 16 \\ 50 & 51 \end{array}\right),\left(\begin{array}{rr} 55 & 18 \\ 54 & 19 \end{array}\right),\left(\begin{array}{rr} 12 & 11 \\ 49 & 51 \end{array}\right)$.
The torsion field $K:=\Q(E[72])$ is a degree-$165888$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/72\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 27 = 3^{3} \) |
| $3$ | additive | $6$ | \( 32 = 2^{5} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 864d consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 864a1, its twist by $-4$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.216.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1119744.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.143327232.1 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.1925877696823296.2 | \(\Z/4\Z\) | not in database |
| $16$ | 16.0.20542695432781824.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | - | - | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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 $0$.