Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3+x^2-28707x+1491009\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3+x^2z-28707xz^2+1491009z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-37204299x+70122588678\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(59, 54)$ | $2.3680909060567117800795222896$ | $\infty$ |
| $(-21, 1454)$ | $0$ | $4$ |
Integral points
\( \left(-21, 1454\right) \), \( \left(-21, -1434\right) \), \( \left(59, 54\right) \), \( \left(59, -114\right) \), \( \left(131, -66\right) \)
Invariants
| Conductor: | $N$ | = | \( 15162 \) | = | $2 \cdot 3 \cdot 7 \cdot 19^{2}$ |
|
| Discriminant: | $\Delta$ | = | $538210900512768$ | = | $2^{12} \cdot 3 \cdot 7^{2} \cdot 19^{7} $ |
|
| j-invariant: | $j$ | = | \( \frac{55611739513}{11440128} \) | = | $2^{-12} \cdot 3^{-1} \cdot 7^{-2} \cdot 11^{3} \cdot 19^{-1} \cdot 347^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5419470827435538223904045796$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.069727593160333592385890863657$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $0.9136325031445056$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.40534819941553$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.3680909060567117800795222896$ |
|
| Real period: | $\Omega$ | ≈ | $0.49215427849853089541113960622$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2^{2} \cdot 3 )\cdot1\cdot2\cdot2^{2} $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $6.9927964277356397579433688927 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 6.992796428 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.492154 \cdot 2.368091 \cdot 96}{4^2} \\ & \approx 6.992796428\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 69120 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 456 = 2^{3} \cdot 3 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 7 & 6 \\ 450 & 451 \end{array}\right),\left(\begin{array}{rr} 68 & 455 \\ 49 & 450 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 449 & 8 \\ 448 & 9 \end{array}\right),\left(\begin{array}{rr} 289 & 288 \\ 70 & 295 \end{array}\right),\left(\begin{array}{rr} 179 & 174 \\ 290 & 59 \end{array}\right),\left(\begin{array}{rr} 160 & 3 \\ 157 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[456])$ is a degree-$189112320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1083 = 3 \cdot 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 2527 = 7 \cdot 19^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 42 = 2 \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 15162.x
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798.e3, its twist by $-19$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.1316928.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.21080517080281344.51 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.15608694214656.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.8.934316546494464.12 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | split | nonsplit | ord | nonsplit | ss | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 3 | 1 | 1 | 1,1 | 3 | 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 | 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.