Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-463562x-122187996\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-463562xz^2-122187996z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-600776379x-5699000812266\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1180, 30622)$ | $0$ | $3$ |
Integral points
\( \left(1180, 30622\right) \), \( \left(1180, -31802\right) \)
Invariants
| Conductor: | $N$ | = | \( 1734 \) | = | $2 \cdot 3 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-70299562860135936$ | = | $-1 \cdot 2^{9} \cdot 3^{9} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{1579268174113}{10077696} \) | = | $-1 \cdot 2^{-9} \cdot 3^{-9} \cdot 7^{3} \cdot 17 \cdot 647^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0698456763578888448107207900$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.18103678032041145797769771142$ |
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| $abc$ quality: | $Q$ | ≈ | $1.037140822363524$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.806568620645966$ | |||
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: | $\Omega$ | ≈ | $0.091448583834496025560690366973$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 243 $ = $ 3^{2}\cdot3^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.4691117635313926901386399083 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.469111764 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.091449 \cdot 1.000000 \cdot 243}{3^2} \\ & \approx 2.469111764\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 38556 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $3$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $17$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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.1.1 | 9.72.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 72 = 2^{3} \cdot 3^{2} \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 55 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 37 & 18 \\ 45 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 55 & 54 \\ 0 & 55 \end{array}\right),\left(\begin{array}{rr} 65 & 54 \\ 63 & 19 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 8 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 9 & 1 \end{array}\right),\left(\begin{array}{rr} 55 & 18 \\ 54 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[72])$ is a degree-$41472$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/72\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 867 = 3 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 289 = 17^{2} \) |
| $17$ | additive | $114$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 1734l
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1734j1, its twist by $17$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.6936.1 | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.867.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.1154594304.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $9$ | 9.1.1001033261568.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.3006202772296403717455872.1 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
| $18$ | 18.0.1539175819415758703337406464.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 17 |
|---|---|---|---|
| Reduction type | split | split | add |
| $\lambda$-invariant(s) | 1 | 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 $0$.