Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+81x+486\)
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(homogenize, simplify) |
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\(y^2z=x^3+81xz^2+486z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+81x+486\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 324 \) | = | $2^{2} \cdot 3^{4}$ |
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| Discriminant: | $\Delta$ | = | $-136048896$ | = | $-1 \cdot 2^{8} \cdot 3^{12} $ |
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| j-invariant: | $j$ | = | \( 432 \) | = | $2^{4} \cdot 3^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.24474712694738962712348786252$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.3159632820940169372165787887$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7737056144690831$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.479624933136263$ | |||
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: | $\Omega$ | ≈ | $1.2835835295820409427940858094$ |
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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)$ | ≈ | $1.2835835295820409427940858094 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.283583530 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.283584 \cdot 1.000000 \cdot 1}{1^2} \\ & \approx 1.283583530\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 108 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 4 | 12 | 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 |
|---|---|---|
| $2$ | 2G | 8.4.0.2 |
| $3$ | 3B.1.2 | 3.8.0.2 |
| $5$ | 5S4 | 5.5.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $160$, genus $4$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 60 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 66 \\ 78 & 49 \end{array}\right),\left(\begin{array}{rr} 88 & 69 \\ 27 & 103 \end{array}\right),\left(\begin{array}{rr} 39 & 20 \\ 110 & 39 \end{array}\right),\left(\begin{array}{rr} 61 & 60 \\ 60 & 61 \end{array}\right),\left(\begin{array}{rr} 109 & 42 \\ 66 & 97 \end{array}\right),\left(\begin{array}{rr} 1 & 60 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 61 & 105 \\ 0 & 91 \end{array}\right),\left(\begin{array}{rr} 61 & 60 \\ 90 & 1 \end{array}\right),\left(\begin{array}{rr} 101 & 20 \\ 100 & 21 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 97 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$221184$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $2$ | \( 81 = 3^{4} \) |
| $3$ | additive | $2$ | \( 4 = 2^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 324d
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-11664.1-b2 |
| $3$ | 3.1.324.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.108.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.419904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.314928.2 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.14693280768.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.180551034077184.5 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.1586874322944.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.1344540538448023869960192.6 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.647677499181836009472.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.3454279995636458717184.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 |
|---|---|---|
| Reduction type | add | add |
| $\lambda$-invariant(s) | - | - |
| $\mu$-invariant(s) | - | - |
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 $0$.