Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-7593771927x-254703853363449\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-7593771927xz^2-254703853363449z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9841528417419x-11883433457939824314\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-13086606881/260100, 3307080065083/132651000)$ | $19.159726693008952451862662194$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 128226 \) | = | $2 \cdot 3 \cdot 7 \cdot 43 \cdot 71$ |
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| Discriminant: | $\Delta$ | = | $153101623358112538524114$ | = | $2 \cdot 3 \cdot 7 \cdot 43^{3} \cdot 71^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{48427980631254958469835824847167473}{153101623358112538524114} \) | = | $2^{-1} \cdot 3^{-1} \cdot 7^{-1} \cdot 43^{-3} \cdot 53^{3} \cdot 71^{-9} \cdot 449^{3} \cdot 523^{3} \cdot 29287^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.0983154773231799158649053814$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $4.0983154773231799158649053814$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0197832790476284$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.79037953917922$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $19.159726693008952451862662194$ |
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| Real period: | $\Omega$ | ≈ | $0.016172798195026061785676212374$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot1\cdot1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.3663926185029717346213588869 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $9$ = $3^2$ (rounded) |
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BSD formula
$$\begin{aligned} 8.366392619 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.016173 \cdot 19.159727 \cdot 3}{1^2} \\ & \approx 8.366392619\end{aligned}$$
Modular invariants
Modular form 128226.2.a.s
For more coefficients, see the Downloads section to the right.
| Modular degree: | 151165440 |
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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 semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $43$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $71$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
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 | 9.72.0.12 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1538712 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 43 \cdot 71 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 769357 & 18 \\ 769365 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1154035 & 769374 \\ 1154043 & 42905 \end{array}\right),\left(\begin{array}{rr} 1365337 & 18 \\ 1517049 & 163 \end{array}\right),\left(\begin{array}{rr} 384679 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 393625 & 18 \\ 465201 & 163 \end{array}\right),\left(\begin{array}{rr} 1538695 & 18 \\ 1538694 & 19 \end{array}\right),\left(\begin{array}{rr} 219817 & 18 \\ 439641 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[1538712])$ is a degree-$6989597150768057548800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1538712\Z)$.
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 | $4$ | \( 64113 = 3 \cdot 7 \cdot 43 \cdot 71 \) |
| $3$ | split multiplicative | $4$ | \( 14 = 2 \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 18318 = 2 \cdot 3 \cdot 43 \cdot 71 \) |
| $43$ | split multiplicative | $44$ | \( 2982 = 2 \cdot 3 \cdot 7 \cdot 71 \) |
| $71$ | nonsplit multiplicative | $72$ | \( 1806 = 2 \cdot 3 \cdot 7 \cdot 43 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 128226ba
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.512904.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.47628.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.6.134929918510539264.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.6805279152.4 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | \(\Q(\zeta_{9})\) | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.6805279152.7 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.789211539648.1 | \(\Z/6\Z\) | not in database |
| $9$ | 9.1.23713062262569607456841614315776.1 | \(\Z/9\Z\) | not in database |
| $9$ | 9.3.8264121852787485710189617152.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.229755206741312113518796221100032.11 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.1686927965605508492466881426820189615618051032124693313927446528.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.204887129993158996908154926247401770109533251512997773312.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.190442057673969415287970766750545528128012288.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.204887129993158996908154926247401770109533251512997773312.2 | \(\Z/18\Z\) | not in database |
| $18$ | 18.6.5223999197625700786764799139517320521212576392467965060117822758191104.1 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 71 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | ord | split | ord | ord | ss | ord | ord | ord | ord | ord | ord | split | ord | nonsplit |
| $\lambda$-invariant(s) | 4 | 4 | 1 | 2 | 1 | 1 | 1,1 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 2 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.