Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-4x+4\)
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(homogenize, simplify) |
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\(y^2z=x^3-4xz^2+4z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4x+4\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2, 2)$ | $0.040264364336880642943215152248$ | $\infty$ |
Integral points
\((-2,\pm 2)\), \((0,\pm 2)\), \((1,\pm 1)\), \((2,\pm 2)\), \((6,\pm 14)\), \((8,\pm 22)\), \((310,\pm 5458)\)
Invariants
| Conductor: | $N$ | = | \( 88 \) | = | $2^{3} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-2816$ | = | $-1 \cdot 2^{8} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( -\frac{27648}{11} \) | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 11^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.62921188472456407340791357381$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0913100050978609463527349881$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7866607319557362$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6396042860016586$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.040264364336880642943215152248$ |
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| Real period: | $\Omega$ | ≈ | $4.2525295331483600253449531458$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $0.68490159390412206930227406664 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 0.684901594 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.252530 \cdot 0.040264 \cdot 4}{1^2} \\ & \approx 0.684901594\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{1}^{*}$ | additive | 1 | 3 | 8 | 0 |
| $11$ | $1$ | $I_{1}$ | 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 label 22.2.0.a.1, level \( 22 = 2 \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 21 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 20 & 3 \end{array}\right),\left(\begin{array}{rr} 13 & 2 \\ 13 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[22])$ is a degree-$39600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22\Z)$.
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 | $2$ | \( 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 8 = 2^{3} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 88a 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 |
|---|---|---|---|
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.32788343808.1 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.20433779818496.1 | \(\Z/4\Z\) | 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 | ss | ord | ord | nonsplit | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 5 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-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.
Additional information
$E$ parametrizes triangles $ABC$ with rational sides $a,b,c$ for which the altitude from $A$, angle bisector to $B$, and median from $C$ are concurrent. Equivalently, $c (a^2 + b^2 - c^2) = a (a^2 + c^2 - b^2)$ [proof by standard triangle geometry, including "Ceva's theorem"]. This elliptic curve is put in standard Weierstrass form by taking $c = ((2/x) - 1) a$, when $y^2 = x^3 - 4x + 4$. The generator $(x,y)=(2,2)$ of the Mordell-Weil group corresponds to an equilateral triangle. The higher multiples that yield positive $(a:b:c)$ are the 7th, 10th, and 12th, with x-coordinates $10/9$, $88/49$, $206/961$, and triangles $(a:b:c) = (15:13:12), (308:277:35), (3193:26447:26598).$ This has been independently observed many times, going back at least to 1939; see [Albime triangles and Guy's favourite elliptic curve] (in Expo. Math. 2015) by Erika Bakker, Jasbir S. Chahal, and Jaap Top.
The title of the Bakker-Chahal-Top paper (and of this knowl) come from Richard Guy's paper "My Favorite Elliptic Curve: A Tale of Two Types of Triangles", Amer. Math. Monthly 102 #9 (Nov. 1995), 771-781, where the same curve also arises in two other contexts; notably, it parametrizes pairs $(R,T)$ of a rectangle $R$ and an isosceles triangle $R$, both with rational sides, and with the same perimeter and area. (For example, $R$ can be a $2 \times 6$ rectangle of perimeter $16$ and area $12$, same as a 5-5-6 triangle $T$ made from two copies of the Pythagorean 3-4-5.)