Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2\)
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(homogenize, simplify) |
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\(y^2z=x^3-2z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3, 5)$ | $1.3495768356801180454777611856$ | $\infty$ |
Integral points
\((3,\pm 5)\)
Invariants
| Conductor: | $N$ | = | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ |
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| Discriminant: | $\Delta$ | = | $-1728$ | = | $-1 \cdot 2^{6} \cdot 3^{3} $ |
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| j-invariant: | $j$ | = | \( 0 \) | = | $0$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-3})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.69989076598103783743244182590$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3211174284280379149898691959$ |
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| $abc$ quality: | $Q$ | ≈ | $$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.0$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3495768356801180454777611856$ |
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| Real period: | $\Omega$ | ≈ | $2.1636817490137510094722028257$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9200547682528014610166866824 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.920054768 \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 2.163682 \cdot 1.349577 \cdot 1}{1^2} \\ & \approx 2.920054768\end{aligned}$$
Modular invariants
\( q + q^{7} - 5 q^{13} - 7 q^{19} + O(q^{20}) \)
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 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 |
|---|---|---|
| $3$ | 3Cs | 27.972.55.16 |
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 | $2$ | \( 27 = 3^{3} \) |
| $3$ | additive | $2$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1728.o
consists of 4 curves linked by isogenies of
degrees dividing 27.
Twists
The minimal quadratic twist of this elliptic curve is 27.a4, its twist by $-8$.
The minimal sextic twist of this elliptic curve is 27.a4, its sextic 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/3\Z\) | 2.2.24.1-81.1-b4 |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/3\Z\) | 2.0.8.1-729.4-b3 |
| $3$ | 3.1.108.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.3359232.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.4478976.4 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.1492992.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.962938848411648.4 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.101559956668416.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | not in database |
| $12$ | 12.0.20061226008576.4 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.543924745232899335442661376.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.1768591357765866863198208.1 | \(\Z/18\Z\) | 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 | add | ss | ord | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | - | 1,1 | 5 | 1,1 | 1 | 1,1 | 3 | 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 | 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.