Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3-x^2-208x+1412\)
|
(homogenize, simplify) |
|
\(y^2z=x^3-x^2z-208xz^2+1412z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-16875x+978750\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-8, 50)$ | $0.19637779133168667685371410477$ | $\infty$ |
Integral points
\((-8,\pm 50)\), \((8,\pm 14)\), \((17,\pm 50)\)
Invariants
| Conductor: | $N$ | = | \( 800 \) | = | $2^{5} \cdot 5^{2}$ |
|
| Discriminant: | $\Delta$ | = | $-200000000$ | = | $-1 \cdot 2^{9} \cdot 5^{8} $ |
|
| j-invariant: | $j$ | = | \( -5000 \) | = | $-1 \cdot 2^{3} \cdot 5^{4}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.31558410008219759058368225303$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2772348936271616412130813935$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9949912704917342$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.177933179061148$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.19637779133168667685371410477$ |
|
| Real period: | $\Omega$ | ≈ | $1.7115587321269884745620801932$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot3 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $2.0166727412973597372967082909 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 2.016672741 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.711559 \cdot 0.196378 \cdot 6}{1^2} \\ & \approx 2.016672741\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 240 |
|
| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
|
Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 5 | 9 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.1 |
| $5$ | 5Ns | 5.15.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 40.60.3.u.1, level \( 40 = 2^{3} \cdot 5 \), index $60$, genus $3$, and generators
$\left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right),\left(\begin{array}{rr} 31 & 32 \\ 8 & 39 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 9 \end{array}\right),\left(\begin{array}{rr} 31 & 10 \\ 20 & 31 \end{array}\right),\left(\begin{array}{rr} 31 & 30 \\ 30 & 11 \end{array}\right),\left(\begin{array}{rr} 21 & 20 \\ 20 & 21 \end{array}\right),\left(\begin{array}{rr} 31 & 0 \\ 0 & 31 \end{array}\right),\left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 31 & 25 \\ 25 & 6 \end{array}\right),\left(\begin{array}{rr} 9 & 8 \\ 15 & 23 \end{array}\right),\left(\begin{array}{rr} 33 & 0 \\ 0 & 33 \end{array}\right),\left(\begin{array}{rr} 31 & 22 \\ 3 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[40])$ is a degree-$12288$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/40\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 25 = 5^{2} \) |
| $5$ | additive | $10$ | \( 32 = 2^{5} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 800.b consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 800.c1, its twist by $-20$.
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 |
|---|---|---|---|
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.89579520000.6 | \(\Z/3\Z\) | not in database |
| $8$ | 8.0.5120000000.3 | \(\Z/5\Z\) | not in database |
| $12$ | 12.2.52428800000000.2 | \(\Z/4\Z\) | not in database |
| $16$ | 16.4.26214400000000000000.1 | \(\Z/5\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 1 | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.