Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-43125x+1030080\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-43125xz^2+1030080z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-55889379x+48227092254\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-117, 2174\right) \) | $1.4312939334493554654113312203$ | $\infty$ |
| \( \left(1470, 55074\right) \) | $4.2115594409451163880043360628$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-117:2174:1]\) | $1.4312939334493554654113312203$ | $\infty$ |
| \([1470:55074:1]\) | $4.2115594409451163880043360628$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-4209, 457056\right) \) | $1.4312939334493554654113312203$ | $\infty$ |
| \( \left(52923, 12054852\right) \) | $4.2115594409451163880043360628$ | $\infty$ |
Integral points
\( \left(-117, 2174\right) \), \( \left(-117, -2058\right) \), \( \left(-21, 1398\right) \), \( \left(-21, -1378\right) \), \( \left(234, 1823\right) \), \( \left(234, -2058\right) \), \( \left(1470, 55074\right) \), \( \left(1470, -56545\right) \)
\([-117:2174:1]\), \([-117:-2058:1]\), \([-21:1398:1]\), \([-21:-1378:1]\), \([234:1823:1]\), \([234:-2058:1]\), \([1470:55074:1]\), \([1470:-56545:1]\)
\((-4209,\pm 457056)\), \((-753,\pm 299808)\), \((8427,\pm 419148)\), \((52923,\pm 12054852)\)
Invariants
| Conductor: | $N$ | = | \( 83582 \) | = | $2 \cdot 23^{2} \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $4671197867306944$ | = | $2^{6} \cdot 23^{6} \cdot 79^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{59914169497}{31554496} \) | = | $2^{-6} \cdot 7^{3} \cdot 13^{3} \cdot 43^{3} \cdot 79^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6982990330123053141789741098$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.13055192504773046877559769390$ |
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| $abc$ quality: | $Q$ | ≈ | $0.967983438848399$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8495454061270116$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.6315188760204778881414814597$ |
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| Real period: | $\Omega$ | ≈ | $0.38117508409973575567365030332$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.5863787247054201416911998237 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.586378725 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \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 0.381175 \cdot 5.631519 \cdot 4}{1^2} \\ & \approx 8.586378725\end{aligned}$$
Modular invariants
Modular form 83582.2.a.j
For more coefficients, see the Downloads section to the right.
| Modular degree: | 475200 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $23$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $79$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 |
|---|---|---|---|
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 65412 = 2^{2} \cdot 3^{2} \cdot 23 \cdot 79 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 65395 & 18 \\ 65394 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 52625 & 22770 \\ 56718 & 59617 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 34127 & 0 \\ 0 & 65411 \end{array}\right),\left(\begin{array}{rr} 11386 & 11385 \\ 21321 & 54028 \end{array}\right),\left(\begin{array}{rr} 42988 & 11385 \\ 63963 & 17044 \end{array}\right)$.
The torsion field $K:=\Q(E[65412])$ is a degree-$26627213482721280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/65412\Z)$.
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 | $4$ | \( 41791 = 23^{2} \cdot 79 \) |
| $3$ | good | $2$ | \( 529 = 23^{2} \) |
| $23$ | additive | $266$ | \( 158 = 2 \cdot 79 \) |
| $79$ | nonsplit multiplicative | $80$ | \( 1058 = 2 \cdot 23^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 83582d
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 158d1, its twist by $-23$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{69}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.3.316.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-23})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.32803594704.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.1214947952.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.6.3324370693321969381621209706770509949814285715951616.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.41234259960336784588724566939339482330761487.1 | \(\Z/9\Z\) | 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 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | ord | ord | ss | ord | ss | ord | add | ss | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 6 | 2 | 2 | 2 | 2,2 | 2 | 2,2 | 2 | - | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | 0 | 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.