Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-15992x-710341\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-15992xz^2-710341z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-255867x-45717674\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-83, 241)$ | $1.2966473205100588265268980002$ | $\infty$ |
| $(-63, 241)$ | $0$ | $6$ |
Integral points
\( \left(-83, 241\right) \), \( \left(-83, -159\right) \), \( \left(-77, 283\right) \), \( \left(-77, -207\right) \), \( \left(-63, 241\right) \), \( \left(-63, -179\right) \), \( \left(147, 241\right) \), \( \left(147, -389\right) \), \( \left(217, 2341\right) \), \( \left(217, -2559\right) \), \( \left(567, 12841\right) \), \( \left(567, -13409\right) \), \( \left(1197, 40561\right) \), \( \left(1197, -41759\right) \)
Invariants
| Conductor: | $N$ | = | \( 8190 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $41294799000000$ | = | $2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 7^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{16751080718799363}{1529437000000} \) | = | $2^{-6} \cdot 3^{3} \cdot 5^{-6} \cdot 7^{-6} \cdot 13^{-1} \cdot 17^{3} \cdot 29^{3} \cdot 173^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3518138738870579455384252194$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0771608017200305226896139102$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9700760822949506$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.51166003764604$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2966473205100588265268980002$ |
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| Real period: | $\Omega$ | ≈ | $0.42702763894844731876510673535$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 432 $ = $ ( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6444509259148926054546993700 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.644450926 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.427028 \cdot 1.296647 \cdot 432}{6^2} \\ & \approx 6.644450926\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 27648 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5460 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 5449 & 12 \\ 5448 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1832 & 11 \\ 1785 & 5428 \end{array}\right),\left(\begin{array}{rr} 3790 & 3 \\ 1233 & 5452 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3186 & 5017 \\ 3185 & 5006 \end{array}\right),\left(\begin{array}{rr} 3277 & 12 \\ 3282 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 5410 & 5451 \end{array}\right),\left(\begin{array}{rr} 2341 & 12 \\ 3126 & 73 \end{array}\right)$.
The torsion field $K:=\Q(E[5460])$ is a degree-$1217325957120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5460\Z)$.
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 | $4$ | \( 39 = 3 \cdot 13 \) |
| $3$ | additive | $6$ | \( 13 \) |
| $5$ | split multiplicative | $6$ | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 8190.bq
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 8190.j1, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{39}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.0.1719900.2 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.62462907.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.7998583451040000.87 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.48214811157963723000000.2 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.10606177828364583359129974337131420048551936000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | split | split | ord | split | ss | ord | ss | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 2 | 4 | 3 | 6 | 1,1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | - | 0 | 0 | 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.