Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+6496x-910519\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+6496xz^2-910519z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+8419437x-42506421138\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(75, -38\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([75:-38:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2703, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(75, -38\right) \)
\([75:-38:1]\)
\( \left(2703, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 82365 \) | = | $3 \cdot 5 \cdot 17^{2} \cdot 19$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-376120651746375$ | = | $-1 \cdot 3^{8} \cdot 5^{3} \cdot 17^{6} \cdot 19 $ |
|
| j-invariant: | $j$ | = | \( \frac{1256216039}{15582375} \) | = | $3^{-8} \cdot 5^{-3} \cdot 13^{3} \cdot 19^{-1} \cdot 83^{3}$ |
|
| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
|
||
| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4775983462950167301398513485$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.060991674266908690015084039561$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9487490541305785$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6194167672402706$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.26304583373526275650524141442$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{3}\cdot1\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L(E,1)$ | ≈ | $1.0521833349410510260209656577 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 1.052183335 \approx L(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.263046 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.052183335\end{aligned}$$
Modular invariants
Modular form 82365.2.a.n
For more coefficients, see the Downloads section to the right.
| Modular degree: | 368640 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $17$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{1}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 4.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 12920 = 2^{3} \cdot 5 \cdot 17 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 12914 & 12915 \end{array}\right),\left(\begin{array}{rr} 7889 & 7888 \\ 2958 & 1055 \end{array}\right),\left(\begin{array}{rr} 4659 & 4658 \\ 9418 & 7515 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 12913 & 8 \\ 12912 & 9 \end{array}\right),\left(\begin{array}{rr} 10639 & 0 \\ 0 & 12919 \end{array}\right),\left(\begin{array}{rr} 10048 & 3043 \\ 9605 & 5322 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 5628 & 5321 \\ 11271 & 766 \end{array}\right)$.
The torsion field $K:=\Q(E[12920])$ is a degree-$148143026995200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12920\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 27455 = 5 \cdot 17^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 5491 = 17^{2} \cdot 19 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 16473 = 3 \cdot 17^{2} \cdot 19 \) |
| $17$ | additive | $146$ | \( 285 = 3 \cdot 5 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 4335 = 3 \cdot 5 \cdot 17^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 82365.n
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 285.c4, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-95}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{85}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-323}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{85}, \sqrt{-95})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.471110640625.4 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 17 | 19 |
|---|---|---|---|---|---|
| Reduction type | ord | split | nonsplit | add | nonsplit |
| $\lambda$-invariant(s) | 9 | 5 | 0 | - | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.