Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+y=x^3-x^2-39433x-3193357\)
|
(homogenize, simplify) |
|
\(y^2z+yz^2=x^3-x^2z-39433xz^2-3193357z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-51105600x-149602518000\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5994910497/595984, 464052285464023/460099648)$ | $19.960647953776460108586177799$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 80275 \) | = | $5^{2} \cdot 13^{2} \cdot 19$ |
|
| Discriminant: | $\Delta$ | = | $-517298170796875$ | = | $-1 \cdot 5^{6} \cdot 13^{6} \cdot 19^{3} $ |
|
| j-invariant: | $j$ | = | \( -\frac{89915392}{6859} \) | = | $-1 \cdot 2^{18} \cdot 7^{3} \cdot 19^{-3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5713266476050815131495956525$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.51586698734273704217752773490$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0331037033479094$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8505007721377655$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $19.960647953776460108586177799$ |
|
| Real period: | $\Omega$ | ≈ | $0.16865743764831545687734929448$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2\cdot1 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.7330234753680576611347989965 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.733023475 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.168657 \cdot 19.960648 \cdot 2}{1^2} \\ & \approx 6.733023475\end{aligned}$$
Modular invariants
Modular form 80275.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 233280 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{3}$ | nonsplit 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 | 9.36.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 66690 = 2 \cdot 3^{3} \cdot 5 \cdot 13 \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 41039 & 0 \\ 0 & 66689 \end{array}\right),\left(\begin{array}{rr} 19 & 54 \\ 59274 & 45613 \end{array}\right),\left(\begin{array}{rr} 43 & 30 \\ 62400 & 63697 \end{array}\right),\left(\begin{array}{rr} 1 & 27 \\ 27 & 730 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 61426 & 54405 \\ 15405 & 26326 \end{array}\right),\left(\begin{array}{rr} 12221 & 11050 \\ 53040 & 60061 \end{array}\right),\left(\begin{array}{rr} 40013 & 0 \\ 0 & 66689 \end{array}\right),\left(\begin{array}{rr} 66637 & 54 \\ 66636 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[66690])$ is a degree-$2258193995366400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/66690\Z)$.
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 |
|---|---|---|---|
| $3$ | good | $2$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
| $5$ | additive | $14$ | \( 3211 = 13^{2} \cdot 19 \) |
| $13$ | additive | $86$ | \( 475 = 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 4225 = 5^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 80275a
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a1, its twist by $65$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{65}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-195}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{65})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1586234000.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.42828318000.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.33418807565459425587973499291015625.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.349572530757794134615698751587893522091796875.1 | \(\Z/9\Z\) | 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 | ss | ord | add | ord | ord | add | ord | nonsplit | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,5 | 3 | - | 1 | 1 | - | 1 | 1 | 1,3 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0,0 | 1 | - | 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.