Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-40x+64\)
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(homogenize, simplify) |
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\(y^2z=x^3-40xz^2+64z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-40x+64\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(0, 8)$ | $0.85551112457175026295371098561$ | $\infty$ |
| $(8, 16)$ | $1.2127710864456289893102859020$ | $\infty$ |
Integral points
\((-7,\pm 1)\), \((0,\pm 8)\), \((1,\pm 5)\), \((8,\pm 16)\), \((168,\pm 2176)\)
Invariants
| Conductor: | $N$ | = | \( 18176 \) | = | $2^{8} \cdot 71$ |
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| Discriminant: | $\Delta$ | = | $2326528$ | = | $2^{15} \cdot 71 $ |
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| j-invariant: | $j$ | = | \( \frac{216000}{71} \) | = | $2^{6} \cdot 3^{3} \cdot 5^{3} \cdot 71^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.075349457675382709225824526391$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.94178343337531434599736467821$ |
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| $abc$ quality: | $Q$ | ≈ | $0.6300092726327309$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.3124562941285243$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0193784251572509792504240589$ |
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| Real period: | $\Omega$ | ≈ | $2.3870376848696088231520834788$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.8665894319867843719285710654 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.866589432 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.387038 \cdot 1.019378 \cdot 2}{1^2} \\ & \approx 4.866589432\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5376 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 8 | 15 | 0 |
| $71$ | $1$ | $I_{1}$ | split 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 \( 568 = 2^{3} \cdot 71 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 433 & 2 \\ 433 & 3 \end{array}\right),\left(\begin{array}{rr} 567 & 2 \\ 566 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 143 & 2 \\ 143 & 3 \end{array}\right),\left(\begin{array}{rr} 285 & 2 \\ 285 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 567 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[568])$ is a degree-$19237478400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/568\Z)$.
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 | $4$ | \( 71 \) |
| $71$ | split multiplicative | $72$ | \( 256 = 2^{8} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 18176j consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 18176o1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.568.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.183250432.1 | \(\Z/2\Z \oplus \Z/2\Z\) | 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 | 71 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ss | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | - | 2,4 | 4,2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 |
| $\mu$-invariant(s) | - | 0,0 | 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.