Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+579x+366533\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+579xz^2+366533z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9261x+23467374\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 643\right) \) | $0.54868622661760549273138168786$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([37:643:1]\) | $0.54868622661760549273138168786$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(147, 5292\right) \) | $0.54868622661760549273138168786$ | $\infty$ |
Integral points
\( \left(-61, 349\right) \), \( \left(-61, -288\right) \), \( \left(37, 643\right) \), \( \left(37, -680\right) \)
\([-61:349:1]\), \([-61:-288:1]\), \([37:643:1]\), \([37:-680:1]\)
\((-245,\pm 2548)\), \((147,\pm 5292)\)
Invariants
| Conductor: | $N$ | = | \( 882 \) | = | $2 \cdot 3^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-58095911978496$ | = | $-1 \cdot 2^{9} \cdot 3^{9} \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{189}{512} \) | = | $2^{-9} \cdot 3^{3} \cdot 7$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3200404410258057951700032255$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.80119220817881867677999919782$ |
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| $abc$ quality: | $Q$ | ≈ | $1.3265882735428125$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.77211945573254$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.54868622661760549273138168786$ |
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| Real period: | $\Omega$ | ≈ | $0.49145006230238338669667269759$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.6179112815340912147969660175 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.617911282 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.491450 \cdot 0.548686 \cdot 6}{1^2} \\ & \approx 1.617911282\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3024 |
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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 |
| $3$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 0 |
| $7$ | $3$ | $IV^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B.1.2 | 9.24.0.4 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 487 & 18 \\ 486 & 19 \end{array}\right),\left(\begin{array}{rr} 238 & 117 \\ 207 & 86 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 243 & 496 \end{array}\right),\left(\begin{array}{rr} 205 & 486 \\ 234 & 265 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 440 & 445 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$83607552$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 | $4$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | additive | $2$ | \( 49 = 7^{2} \) |
| $7$ | additive | $26$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 882a
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 882b2, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-86436.3-j2 |
| $3$ | 3.1.1176.1 | \(\Z/2\Z\) | not in database |
| $3$ | \(\Q(\sqrt[3]{7})\) | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.33191424.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.5250987.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.4148928.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.3556892570112.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.1101670627147776.5 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.8549394874383196572862347.3 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.37954454265953846507077632.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.19432680584168369411623747584.5 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 1 | - | 1 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.