Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x+9\)
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(homogenize, simplify) |
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\(y^2z=x^3-xz^2+9z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-x+9\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1, 3)$ | $0.75796330945717768686987863081$ | $\infty$ |
| $(-1, 3)$ | $0.77584813338166073629441238515$ | $\infty$ |
Integral points
\((-1,\pm 3)\), \((0,\pm 3)\), \((1,\pm 3)\), \((9,\pm 27)\), \((35,\pm 207)\), \((37,\pm 225)\), \((46584,\pm 10054377)\)
Invariants
| Conductor: | $N$ | = | \( 8732 \) | = | $2^{2} \cdot 37 \cdot 59$ |
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| Discriminant: | $\Delta$ | = | $-34928$ | = | $-1 \cdot 2^{4} \cdot 37 \cdot 59 $ |
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| j-invariant: | $j$ | = | \( -\frac{6912}{2183} \) | = | $-1 \cdot 2^{8} \cdot 3^{3} \cdot 37^{-1} \cdot 59^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.44906375959669217037615658674$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.68011281978334060684856729389$ |
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| $abc$ quality: | $Q$ | ≈ | $0.811193223149947$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $1.974445053552544$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.55925381077285617072644253823$ |
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| Real period: | $\Omega$ | ≈ | $2.9861792618199655152486339109$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 3\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.0100963954710709535929150965 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.010096395 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.986179 \cdot 0.559254 \cdot 3}{1^2} \\ & \approx 5.010096395\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 456 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $37$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $59$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4366 = 2 \cdot 37 \cdot 59 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 4365 & 2 \\ 4364 & 3 \end{array}\right),\left(\begin{array}{rr} 297 & 2 \\ 297 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1889 & 2 \\ 1889 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 4365 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[4366])$ is a degree-$65098476679680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4366\Z)$.
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 | $2$ | \( 2183 = 37 \cdot 59 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 236 = 2^{2} \cdot 59 \) |
| $59$ | nonsplit multiplicative | $60$ | \( 148 = 2^{2} \cdot 37 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 8732.b consists of this curve only.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.2183.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.10403062487.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | - | 4,2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 0,0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.