Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-270x-1708\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-270xz^2-1708z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4320x-109296\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 27 \) | = | $3^{3}$ |
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| Discriminant: | $\Delta$ | = | $-177147$ | = | $-1 \cdot 3^{11} $ |
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| j-invariant: | $j$ | = | \( -12288000 \) | = | $-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-27})/2]\) (potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $0.052147932407099199254187350298$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.95491333220533468452478745021$ |
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| $abc$ quality: | $Q$ | ≈ | $1.23864473399791$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.619664953039187$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.58887958342848331910456316655$ |
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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)$ | ≈ | $0.58887958342848331910456316655 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.588879583 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.588880 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 0.588879583\end{aligned}$$
Modular invariants
\( q - 2 q^{4} - q^{7} + 5 q^{13} + 4 q^{16} - 7 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
| Modular degree: | 3 |
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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 is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $II^{*}$ | additive | -1 | 3 | 11 | 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 |
|---|---|---|
| $3$ | 3B.1.2 | 27.648.13.34 |
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 |
|---|---|---|---|
| $3$ | additive | $4$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 9 and 27.
Its isogeny class 27a
consists of 4 curves linked by isogenies of
degrees dividing 27.
Twists
The minimal quadratic twist of this elliptic curve is 27a4, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-81.1-CMa2 |
| $3$ | 3.1.108.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.243.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | \(\Q(\zeta_{9})\) | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.177147.1 | \(\Z/9\Z\) | not in database |
| $9$ | 9.1.24794911296.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.15045919506432.1 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.241162079949.1 | \(\Z/21\Z\) | not in database |
| $18$ | 18.0.4052555153018976267.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.2954312706550833698643.2 | \(\Z/27\Z\) | not in database |
| $18$ | \(\Q(\zeta_{27})\) | \(\Z/27\Z\) | not in database |
| $18$ | 18.0.1844362878529525198848.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.2529990231179046912.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $18$ | 18.0.1844362878529525198848.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 |
|---|---|---|
| Reduction type | ss | add |
| $\lambda$-invariant(s) | 0,5 | - |
| $\mu$-invariant(s) | 0,0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.