Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2-883x+16613\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z-883xz^2+16613z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-71550x+12325500\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-7, 150)$ | $0.56233353028951204966441749244$ | $\infty$ |
| $(-37, 0)$ | $0$ | $2$ |
Integral points
\( \left(-37, 0\right) \), \((-7,\pm 150)\), \((19,\pm 84)\), \((68,\pm 525)\), \((453563,\pm 305461800)\)
Invariants
| Conductor: | $N$ | = | \( 67200 \) | = | $2^{7} \cdot 3 \cdot 5^{2} \cdot 7$ |
|
| Discriminant: | $\Delta$ | = | $-79380000000$ | = | $-1 \cdot 2^{8} \cdot 3^{4} \cdot 5^{7} \cdot 7^{2} $ |
|
| j-invariant: | $j$ | = | \( -\frac{19056256}{19845} \) | = | $-1 \cdot 2^{7} \cdot 3^{-4} \cdot 5^{-1} \cdot 7^{-2} \cdot 53^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.78673055003519707619285277725$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.48008652655514998405234830367$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.8450078970255808$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.968320986000666$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.56233353028951204966441749244$ |
|
| Real period: | $\Omega$ | ≈ | $0.98632369343628111281995597037$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $8.8742861526114301178190465008 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 8.874286153 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.986324 \cdot 0.562334 \cdot 64}{2^2} \\ & \approx 8.874286153\end{aligned}$$
Modular invariants
Modular form 67200.2.a.fx
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49152 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III$ | additive | 1 | 7 | 8 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 280 = 2^{3} \cdot 5 \cdot 7 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 36 & 249 \\ 177 & 114 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 114 & 1 \\ 223 & 0 \end{array}\right),\left(\begin{array}{rr} 241 & 4 \\ 202 & 9 \end{array}\right),\left(\begin{array}{rr} 277 & 4 \\ 276 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 141 & 4 \\ 2 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[280])$ is a degree-$123863040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/280\Z)$.
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 | $2$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 22400 = 2^{7} \cdot 5^{2} \cdot 7 \) |
| $5$ | additive | $18$ | \( 2688 = 2^{7} \cdot 3 \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 9600 = 2^{7} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 67200.fx
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 13440.l2, its twist by $-40$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.250880.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.802816000000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6294077440000.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | split | add | split | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 1,1 | 1 | 1 | 3 | 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.