Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1271x+1105962\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1271xz^2+1105962z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20339x+70761230\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4421}{16}, \frac{289933}{64}\right) \) | $6.4989813549396113111719292446$ | $\infty$ |
| \( \left(-\frac{429}{4}, \frac{429}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([17684:289933:64]\) | $6.4989813549396113111719292446$ | $\infty$ |
| \([-858:429:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{4417}{4}, \frac{298775}{8}\right) \) | $6.4989813549396113111719292446$ | $\infty$ |
| \( \left(-430, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 31433 \) | = | $17 \cdot 43^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-527966563215529$ | = | $-1 \cdot 17^{4} \cdot 43^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{35937}{83521} \) | = | $-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5039639369054377307789882524$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.37663612094134348095743300427$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1807067659885138$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9934884291713284$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.4989813549396113111719292446$ |
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| Real period: | $\Omega$ | ≈ | $0.41872129001499420984877423109$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4425237134474180211034744837 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.442523713 \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 0.418721 \cdot 6.498981 \cdot 8}{2^2} \\ & \approx 5.442523713\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 80640 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $43$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | 16.48.0.76 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 46784 = 2^{6} \cdot 17 \cdot 43 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 9 & 124 \\ 22516 & 39913 \end{array}\right),\left(\begin{array}{rr} 1 & 64 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 46721 & 64 \\ 46720 & 65 \end{array}\right),\left(\begin{array}{rr} 36078 & 15007 \\ 7267 & 20082 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 64 & 1 \end{array}\right),\left(\begin{array}{rr} 25023 & 0 \\ 0 & 46783 \end{array}\right),\left(\begin{array}{rr} 38271 & 39646 \\ 23822 & 16599 \end{array}\right),\left(\begin{array}{rr} 44033 & 9804 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 57 & 16 \\ 40432 & 45001 \end{array}\right)$.
The torsion field $K:=\Q(E[46784])$ is a degree-$1070880604028928$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/46784\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 1849 = 43^{2} \) |
| $17$ | split multiplicative | $18$ | \( 1849 = 43^{2} \) |
| $43$ | additive | $926$ | \( 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 31433b
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 17a1, its twist by $-43$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{43}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-43}) \) | \(\Z/4\Z\) | 2.0.43.1-289.2-a1 |
| $4$ | \(\Q(i, \sqrt{43})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.224054542336.12 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.18713259430445056.10 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.4568666853136.3 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | ss | ord | ord | ss | ord | split | ord | ord | ord | ord | ord | ord | add | ss |
| $\lambda$-invariant(s) | 4 | 1,1 | 1 | 1 | 1,1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 3 | - | 1,3 |
| $\mu$-invariant(s) | 1 | 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.