Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+32059597x+4539299987\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+32059597xz^2+4539299987z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+512953557x+291028152742\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-77, 45538)$ | $1.0112571244941598497439664269$ | $\infty$ |
| $(2107, 284322)$ | $0$ | $3$ |
Integral points
\( \left(-77, 45538\right) \), \( \left(-77, -45462\right) \), \( \left(2107, 284322\right) \), \( \left(2107, -286430\right) \), \( \left(4165, 456508\right) \), \( \left(4165, -460674\right) \), \( \left(20923, 3125538\right) \), \( \left(20923, -3146462\right) \), \( \left(124411, 43865314\right) \), \( \left(124411, -43989726\right) \)
Invariants
| Conductor: | $N$ | = | \( 8190 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-2117824071863205888000000$ | = | $-1 \cdot 2^{21} \cdot 3^{6} \cdot 5^{6} \cdot 7^{9} \cdot 13^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{4998853083179567995470359}{2905108466204672000000} \) | = | $2^{-21} \cdot 5^{-6} \cdot 7^{-9} \cdot 13^{-3} \cdot 53^{3} \cdot 79^{3} \cdot 97^{3} \cdot 421^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3571477774718163951485493433$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8078416331377615494509267248$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0818447636819732$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.043086725422254$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0112571244941598497439664269$ |
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| Real period: | $\Omega$ | ≈ | $0.049687628279901124764187868175$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1134 $ = $ ( 3 \cdot 7 )\cdot1\cdot2\cdot3^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.3311179802557061791493026222 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.331117980 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.049688 \cdot 1.011257 \cdot 1134}{3^2} \\ & \approx 6.331117980\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1905120 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $21$ | $I_{21}$ | split multiplicative | -1 | 1 | 21 | 21 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $13$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3Cs.1.1 | 3.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 10 & 9 \\ 3267 & 6544 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 1516 & 9 \\ 1503 & 6532 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3754 & 9 \\ 1863 & 6544 \end{array}\right),\left(\begin{array}{rr} 1639 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4903 & 3258 \\ 3438 & 2267 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 6535 & 18 \\ 6534 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[6552])$ is a degree-$2191186722816$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6552\Z)$.
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 | $4$ | \( 819 = 3^{2} \cdot 7 \cdot 13 \) |
| $3$ | additive | $2$ | \( 1 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 8190bl
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 910e2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $3$ | 3.1.728.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.385828352.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.14309568.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.492511526739648.3 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.111957082389109746826171875.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.36807302570912616483000000000000.1 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.727702811914257020738325491712.1 | \(\Z/3\Z \oplus \Z/9\Z\) | 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 | nonsplit | split | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 1 | 4 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 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.