Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1143435x+440919866\)
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(homogenize, simplify) |
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\(y^2z=x^3-1143435xz^2+440919866z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1143435x+440919866\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(455, 3854)$ | $5.8425680263465223041000504224$ | $\infty$ |
| $(-1226, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1226, 0\right) \), \((455,\pm 3854)\)
Invariants
| Conductor: | $N$ | = | \( 19152 \) | = | $2^{4} \cdot 3^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $11693251603455787008$ | = | $2^{14} \cdot 3^{24} \cdot 7 \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{55369510069623625}{3916046302812} \) | = | $2^{-2} \cdot 3^{-18} \cdot 5^{3} \cdot 7^{-1} \cdot 19^{-2} \cdot 31^{3} \cdot 2459^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4060742722258224698277028465$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1636209473318223147128481066$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9950582420479482$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.422045099927932$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.8425680263465223041000504224$ |
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| Real period: | $\Omega$ | ≈ | $0.22176461734720992653444641619$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1826994507511201730039472569 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.182699451 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.221765 \cdot 5.842568 \cdot 16}{2^2} \\ & \approx 5.182699451\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 442368 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $3$ | $4$ | $I_{18}^{*}$ | additive | -1 | 2 | 24 | 18 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 2056 & 9 \\ 3499 & 178 \end{array}\right),\left(\begin{array}{rr} 1285 & 36 \\ 786 & 265 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 1351 \end{array}\right),\left(\begin{array}{rr} 4753 & 36 \\ 4752 & 37 \end{array}\right),\left(\begin{array}{rr} 4778 & 4761 \\ 3137 & 2006 \end{array}\right),\left(\begin{array}{rr} 4255 & 4752 \\ 4246 & 4427 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[4788])$ is a degree-$107226685440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4788\Z)$.
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 | $2$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $2$ | \( 2128 = 2^{4} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 19152bl
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 798e2, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.90972.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.720923261184.4 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.540692445888.2 | \(\Z/18\Z\) | not in database |
| $8$ | 8.0.6488309350656.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.880684648704.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.132414476544.2 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.12.14325067730977042597318656.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | add | add | ss | nonsplit | ord | ord | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1,1 | 1 | 1 | 1 | 1,1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.