Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-32792836x-72282118834\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-32792836xz^2-72282118834z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-42499515483x-3372267037772682\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{52979}{16}, \frac{84283}{64}\right) \) | $3.4629487283130357633637508309$ | $\infty$ |
| \( \left(-\frac{13249}{4}, \frac{13249}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-211916:84283:64]\) | $3.4629487283130357633637508309$ | $\infty$ |
| \([-26498:13249:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{476799}{4}, -\frac{585225}{8}\right) \) | $3.4629487283130357633637508309$ | $\infty$ |
| \( \left(-119238, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 8670 \) | = | $2 \cdot 3 \cdot 5 \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $22079667965176541250$ | = | $2 \cdot 3^{16} \cdot 5^{4} \cdot 17^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{161572377633716256481}{914742821250} \) | = | $2^{-1} \cdot 3^{-16} \cdot 5^{-4} \cdot 17^{-1} \cdot 23^{3} \cdot 236807^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9033465489412446850164223181$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4867398769131366448916550092$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0337884894740492$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.0063298224935915$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.4629487283130357633637508309$ |
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| Real period: | $\Omega$ | ≈ | $0.063089190917473862211367813269$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 1\cdot2^{4}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.9911882706548619890226010306 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.991188271 \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 0.063089 \cdot 3.462949 \cdot 128}{2^2} \\ & \approx 6.991188271\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 589824 |
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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 4 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 |
| $3$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $5$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.96.0.242 | $96$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8160 = 2^{5} \cdot 3 \cdot 5 \cdot 17 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 5441 & 32 \\ 5456 & 513 \end{array}\right),\left(\begin{array}{rr} 2858 & 8157 \\ 3475 & 212 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 3074 & 31 \\ 1949 & 5700 \end{array}\right),\left(\begin{array}{rr} 511 & 32 \\ 6630 & 1 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 5598 & 6155 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 8129 & 32 \\ 8128 & 33 \end{array}\right),\left(\begin{array}{rr} 6551 & 26 \\ 7254 & 5291 \end{array}\right)$.
The torsion field $K:=\Q(E[8160])$ is a degree-$924089057280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8160\Z)$.
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 | $4$ | \( 289 = 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 8670v
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 510e7, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{34}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.4.101240302206976.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.101240302206976.13 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.13684080640000.52 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.253100755517440000.11 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | split | nonsplit | ss | ord | ord | add | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 8 | 2 | 1 | 1,1 | 1 | 1 | - | 1 | 1,1 | 1 | 1 | 3 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 3 | 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.