Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-339x-4555\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-339xz^2-4555z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5427x-296946\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 810 \) | = | $2 \cdot 3^{4} \cdot 5$ |
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| Discriminant: | $\Delta$ | = | $-6802444800$ | = | $-1 \cdot 2^{9} \cdot 3^{12} \cdot 5^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{8120601}{12800} \) | = | $-1 \cdot 2^{-9} \cdot 3^{3} \cdot 5^{-2} \cdot 67^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.57798894725964709629991249531$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.52062334140846259509533274161$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9934255160870687$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.540515907862993$ | |||
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: | $\Omega$ | ≈ | $0.52714928365676209031752440183$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0542985673135241806350488037 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.054298567 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.527149 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 1.054298567\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 648 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 4 | 12 | 0 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 8.2.0.1 |
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-24.a.1.6, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 18 & 23 \\ 23 & 21 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 21 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 81 = 3^{4} \) |
| $3$ | additive | $2$ | \( 5 \) |
| $5$ | split multiplicative | $6$ | \( 162 = 2 \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 810b
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-72900.1-a1 |
| $3$ | 3.1.648.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.675.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.1366875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.1259712.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.114791256000000.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.101559956668416.6 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.128225377888491045948046875.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.39531097362172608000000000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.6746640616477458432000000000000.3 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 |
|---|---|---|---|
| Reduction type | nonsplit | add | split |
| $\lambda$-invariant(s) | 5 | - | 1 |
| $\mu$-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 $0$.