Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy+y=x^3-168479x+26272946\)
|
(homogenize, simplify) |
|
\(y^2z+xyz+yz^2=x^3-168479xz^2+26272946z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-218348163x+1226445624702\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(186, 1081\right) \) | $1.0176780620075371191330489860$ | $\infty$ |
| \( \left(215, -108\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([186:1081:1]\) | $1.0176780620075371191330489860$ | $\infty$ |
| \([215:-108:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6699, 253692\right) \) | $1.0176780620075371191330489860$ | $\infty$ |
| \( \left(7743, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(186, 1081\right) \), \( \left(186, -1268\right) \), \( \left(215, -108\right) \), \( \left(384, 4078\right) \), \( \left(384, -4463\right) \), \( \left(1239, 40852\right) \), \( \left(1239, -42092\right) \)
\([186:1081:1]\), \([186:-1268:1]\), \([215:-108:1]\), \([384:4078:1]\), \([384:-4463:1]\), \([1239:40852:1]\), \([1239:-42092:1]\)
\((6699,\pm 253692)\), \( \left(7743, 0\right) \), \((13827,\pm 922428)\), \((44607,\pm 8957952)\)
Invariants
| Conductor: | $N$ | = | \( 25230 \) | = | $2 \cdot 3 \cdot 5 \cdot 29^{2}$ |
|
| Minimal Discriminant: | $\Delta$ | = | $7550512980295680$ | = | $2^{20} \cdot 3^{10} \cdot 5 \cdot 29^{3} $ |
|
| j-invariant: | $j$ | = | \( \frac{21685195471991381}{309586821120} \) | = | $2^{-20} \cdot 3^{-10} \cdot 5^{-1} \cdot 11^{3} \cdot 101^{3} \cdot 251^{3}$ |
|
| 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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8510896537505201232424851369$ |
|
||
| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0092656962539016164466671288$ |
|
||
| $abc$ quality: | $Q$ | ≈ | $1.0328645517916797$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.707802567739306$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
|
| Mordell-Weil rank: | $r$ | = | $ 1$ |
|
| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0176780620075371191330489860$ |
|
| Real period: | $\Omega$ | ≈ | $0.41839367381360438903757538660$ |
|
| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot( 2 \cdot 5 )\cdot1\cdot2 $ |
|
| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
|
| Special value: | $ L'(E,1)$ | ≈ | $4.2579006312284254683450093876 $ |
|
| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
|
BSD formula
$$\begin{aligned} 4.257900631 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.418394 \cdot 1.017678 \cdot 40}{2^2} \\ & \approx 4.257900631\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 224000 |
|
| $ \Gamma_0(N) $-optimal: | no | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{20}$ | nonsplit multiplicative | 1 | 1 | 20 | 20 |
| $3$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $5$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $2$ | $III$ | additive | -1 | 2 | 3 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $5$ | 5B | 5.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 580 = 2^{2} \cdot 5 \cdot 29 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 62 & 15 \\ 365 & 318 \end{array}\right),\left(\begin{array}{rr} 561 & 20 \\ 560 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 340 & 231 \end{array}\right),\left(\begin{array}{rr} 291 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 122 & 13 \\ 495 & 396 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[580])$ is a degree-$109132800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/580\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 145 = 5 \cdot 29 \) |
| $3$ | split multiplicative | $4$ | \( 8410 = 2 \cdot 5 \cdot 29^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 841 = 29^{2} \) |
| $29$ | additive | $226$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 25230.h
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{145}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-29 -4 \sqrt{-29}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{290 +2 \sqrt{145}})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.0.95171731360000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $20$ | 20.0.125571223144625902073297766037285327911376953125.1 | \(\Z/10\Z\) | 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 | nonsplit | split | nonsplit | ord | ss | ord | ord | ord | ord | add | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | 2 | 1 | 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 | 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.