Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2=x^3+x^2+24703x+579807\)
|
(homogenize, simplify) |
|
\(y^2z=x^3+x^2z+24703xz^2+579807z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3+2000916x+416676528\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(139, 2592\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([139:2592:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1254, 69984\right) \) | $0$ | $4$ |
Integral points
\( \left(-23, 0\right) \), \((139,\pm 2592)\)
\([-23:0:1]\), \([139:\pm 2592:1]\)
\( \left(-23, 0\right) \), \((139,\pm 2592)\)
Invariants
| Conductor: | $N$ | = | \( 1344 \) | = | $2^{6} \cdot 3 \cdot 7$ |
|
| Minimal Discriminant: | $\Delta$ | = | $-1105875083722752$ | = | $-1 \cdot 2^{19} \cdot 3^{16} \cdot 7^{2} $ |
|
| j-invariant: | $j$ | = | \( \frac{6359387729183}{4218578658} \) | = | $2^{-1} \cdot 3^{-16} \cdot 7^{-2} \cdot 97^{3} \cdot 191^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5753955545376111276759352974$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.53567478369769316355008711521$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.083141850233532$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.824689773042287$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
|
| Mordell-Weil rank: | $r$ | = | $ 0$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
|
| Real period: | $\Omega$ | ≈ | $0.30718905810669331026149980802$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{4}\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
|
| Special value: | $ L(E,1)$ | ≈ | $2.4575124648535464820919984642 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
|
BSD formula
$$\begin{aligned} 2.457512465 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.307189 \cdot 1.000000 \cdot 128}{4^2} \\ & \approx 2.457512465\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6144 |
|
| $ \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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $7$ | $2$ | $I_{2}$ | nonsplit 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 16.96.0.111 | $96$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 112 = 2^{4} \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22 & 89 \\ 101 & 58 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 14 & 99 \end{array}\right),\left(\begin{array}{rr} 109 & 16 \\ 16 & 21 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 108 & 109 \end{array}\right),\left(\begin{array}{rr} 20 & 23 \\ 53 & 54 \end{array}\right),\left(\begin{array}{rr} 97 & 16 \\ 96 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[112])$ is a degree-$258048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/112\Z)$.
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 | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 448 = 2^{6} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 1344.q
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 42.a6, its twist by $8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.8.1-7056.2-f2 |
| $4$ | \(\Q(\sqrt[4]{2})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.40282095616.10 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.16777216.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.105226698752.2 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.493455671296.2 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.2.6968605851648.18 | \(\Z/12\Z\) | not in database |
| $16$ | 16.0.1622647227216566419456.13 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 7 |
|---|---|---|---|
| Reduction type | add | split | nonsplit |
| $\lambda$-invariant(s) | - | 3 | 0 |
| $\mu$-invariant(s) | - | 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 $0$.