Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-9x+9\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-9xz^2+9z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-756x+4320\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(5, 8\right) \) | $0.67580186720604166614690603598$ | $\infty$ |
| \( \left(1, 0\right) \) | $0$ | $2$ |
| \( \left(3, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([5:8:1]\) | $0.67580186720604166614690603598$ | $\infty$ |
| \([1:0:1]\) | $0$ | $2$ |
| \([3:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(42, 216\right) \) | $0.67580186720604166614690603598$ | $\infty$ |
| \( \left(6, 0\right) \) | $0$ | $2$ |
| \( \left(24, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-3, 0\right) \), \((-1,\pm 4)\), \((0,\pm 3)\), \( \left(1, 0\right) \), \( \left(3, 0\right) \), \((5,\pm 8)\), \((9,\pm 24)\), \((51,\pm 360)\)
\([-3:0:1]\), \([-1:\pm 4:1]\), \([0:\pm 3:1]\), \([1:0:1]\), \([3:0:1]\), \([5:\pm 8:1]\), \([9:\pm 24:1]\), \([51:\pm 360:1]\)
\( \left(-3, 0\right) \), \((-1,\pm 4)\), \((0,\pm 3)\), \( \left(1, 0\right) \), \( \left(3, 0\right) \), \((5,\pm 8)\), \((9,\pm 24)\), \((51,\pm 360)\)
Invariants
| Conductor: | $N$ | = | \( 192 \) | = | $2^{6} \cdot 3$ |
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| Minimal Discriminant: | $\Delta$ | = | $36864$ | = | $2^{12} \cdot 3^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{21952}{9} \) | = | $2^{6} \cdot 3^{-2} \cdot 7^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.42622037050316614838561476179$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1193675510631114578028468832$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0917548251330267$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4834799464942217$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.67580186720604166614690603598$ |
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| Real period: | $\Omega$ | ≈ | $3.3132763404731883328969367459$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.1195591687306906582052933086 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.119559169 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.313276 \cdot 0.675802 \cdot 8}{4^2} \\ & \approx 1.119559169\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 6 | 12 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 |
|---|---|---|---|
| $2$ | 2Cs | 8.24.0.4 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.48.0-24.f.1.8, level \( 24 = 2^{3} \cdot 3 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 21 & 4 \\ 20 & 5 \end{array}\right),\left(\begin{array}{rr} 3 & 22 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 9 & 22 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 4 \\ 10 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$1536$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 192.a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 96.a3, its twist by $-8$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.8.1-288.2-a4 |
| $4$ | \(\Q(\zeta_{24})^+\) | \(\Z/2\Z \oplus \Z/4\Z\) | 4.4.2304.1-576.1-c3 |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(\zeta_{24})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.603979776.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.46438023168.8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.8.37396835774521933824.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.9573589958277615058944.57 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.