Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-3427962189x+82858170304336\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-3427962189xz^2+82858170304336z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4442638996323x+3865844121636101022\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(18644, 5033292)$ | $0$ | $3$ |
Integral points
\( \left(18644, 5033292\right) \), \( \left(18644, -5051937\right) \)
Invariants
| Conductor: | $N$ | = | \( 25410 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-387875799586243516070840625000$ | = | $-1 \cdot 2^{3} \cdot 3^{15} \cdot 5^{8} \cdot 7^{9} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{20782141595587068688417129}{1809469231117340625000} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-15} \cdot 5^{-8} \cdot 7^{-9} \cdot 11 \cdot 43^{3} \cdot 199^{3} \cdot 14447^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $4.4217264693251612898635455738$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.8231296207929142604889165218$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0510900439832034$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.652591477303491$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.029404231154299819210310825086$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 810 $ = $ 1\cdot( 3 \cdot 5 )\cdot2\cdot3^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6463808038869837289279742577 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.646380804 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.029404 \cdot 1.000000 \cdot 810}{3^2} \\ & \approx 2.646380804\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 47900160 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $15$ | $I_{15}$ | split multiplicative | -1 | 1 | 15 | 15 |
| $5$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $11$ | $3$ | $IV^{*}$ | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 165 & 166 \\ 158 & 161 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 163 & 6 \\ 162 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 78 & 97 \\ 115 & 54 \end{array}\right),\left(\begin{array}{rr} 127 & 6 \\ 45 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 85 & 6 \\ 87 & 19 \end{array}\right),\left(\begin{array}{rr} 73 & 6 \\ 51 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$9289728$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 | $4$ | \( 2541 = 3 \cdot 7 \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 605 = 5 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 1694 = 2 \cdot 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
| $11$ | additive | $52$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 25410.bc
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 25410.ck1, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.20328.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.69422234112.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.247066875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.15887477521141875.6 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.465127190624482910048448000000000000.3 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit | split | add |
| $\lambda$-invariant(s) | 2 | 3 | 2 | 1 | - |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ 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 $0$.