Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2+190049x-54816359\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z+190049xz^2-54816359z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+246302829x-2561206591314\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(232829/400, 125030227/8000)$ | $9.4177691709293314746120404029$ | $\infty$ |
| $(907/4, -907/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 85782 \) | = | $2 \cdot 3 \cdot 17 \cdot 29^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1741153160949869988$ | = | $-1 \cdot 2^{2} \cdot 3^{16} \cdot 17 \cdot 29^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{1276229915423}{2927177028} \) | = | $2^{-2} \cdot 3^{-16} \cdot 17^{-1} \cdot 10847^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1841526103547366131208470121$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.50050469536149959952921099592$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0301007930420596$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.328073598142169$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.4177691709293314746120404029$ |
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| Real period: | $\Omega$ | ≈ | $0.13738464659745748000302083127$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5877137785691126635396205812 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.587713779 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.137385 \cdot 9.417769 \cdot 8}{2^2} \\ & \approx 2.587713779\end{aligned}$$
Modular invariants
Modular form 85782.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1433600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{16}$ | nonsplit multiplicative | 1 | 1 | 16 | 16 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 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 |
|---|---|---|
| $2$ | 2B | 16.48.0.132 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 7888 = 2^{4} \cdot 17 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 5192 & 5713 \\ 2175 & 4090 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5711 & 0 \\ 0 & 7887 \end{array}\right),\left(\begin{array}{rr} 7873 & 16 \\ 7872 & 17 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 7790 & 7875 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 5453 & 4640 \\ 3596 & 7193 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 7884 & 7885 \end{array}\right),\left(\begin{array}{rr} 1538 & 6467 \\ 7743 & 5598 \end{array}\right)$.
The torsion field $K:=\Q(E[7888])$ is a degree-$6839221616640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/7888\Z)$.
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 | $4$ | \( 14297 = 17 \cdot 29^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 28594 = 2 \cdot 17 \cdot 29^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 5046 = 2 \cdot 3 \cdot 29^{2} \) |
| $29$ | additive | $422$ | \( 102 = 2 \cdot 3 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 85782.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 102.c6, its twist by $29$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{29}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-493}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-17}, \sqrt{29})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{29}, \sqrt{34})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{29})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.4370443248611584.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.3871396095655936.46 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | \(\Z/12\Z\) | 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 | nonsplit | ord | ss | ord | ord | nonsplit | ord | ss | add | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | 1 | 3 | 1,1 | 1 | 1 | 1 | 3 | 1,1 | - | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 2 | 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.