Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-3994x+27476\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-3994xz^2+27476z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-5176899x+1359570366\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-20, 326\right) \) | $0.37020301253936637249609312433$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-20:326:1]\) | $0.37020301253936637249609312433$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-705, 68256\right) \) | $0.37020301253936637249609312433$ | $\infty$ |
Integral points
\( \left(-28, 358\right) \), \( \left(-28, -330\right) \), \( \left(-20, 326\right) \), \( \left(-20, -306\right) \), \( \left(59, 10\right) \), \( \left(59, -69\right) \), \( \left(6853, 563912\right) \), \( \left(6853, -570765\right) \)
\([-28:358:1]\), \([-28:-330:1]\), \([-20:326:1]\), \([-20:-306:1]\), \([59:10:1]\), \([59:-69:1]\), \([6853:563912:1]\), \([6853:-570765:1]\)
\((-993,\pm 74304)\), \((-705,\pm 68256)\), \((2139,\pm 8532)\), \((246723,\pm 122545116)\)
Invariants
| Conductor: | $N$ | = | \( 7742 \) | = | $2 \cdot 7^{2} \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $3712354899904$ | = | $2^{6} \cdot 7^{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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1035069995753871213282740656$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.13055192504773046877559769388$ |
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| $abc$ quality: | $Q$ | ≈ | $0.967983438848399$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.075267721173161$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.37020301253936637249609312433$ |
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| Real period: | $\Omega$ | ≈ | $0.69093851586955202830007333230$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.5347251203263217653387456236 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.534725120 \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 0.690939 \cdot 0.370203 \cdot 6}{1^2} \\ & \approx 1.534725120\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15120 |
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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 |
| $7$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $79$ | $3$ | $I_{3}$ | split 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 \( 19908 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 79 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 15653 & 14238 \\ 11214 & 14113 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19891 & 18 \\ 19890 & 19 \end{array}\right),\left(\begin{array}{rr} 9955 & 14238 \\ 7119 & 8695 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4621 & 14238 \\ 19593 & 11887 \end{array}\right),\left(\begin{array}{rr} 17063 & 0 \\ 0 & 19907 \end{array}\right)$.
The torsion field $K:=\Q(E[19908])$ is a degree-$200923996815360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19908\Z)$.
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 | $4$ | \( 3871 = 7^{2} \cdot 79 \) |
| $3$ | good | $2$ | \( 49 = 7^{2} \) |
| $7$ | additive | $26$ | \( 158 = 2 \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 98 = 2 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 7742.b
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 158.b2, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/3\Z\) | 2.0.7.1-24964.5-b1 |
| $3$ | 3.3.316.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-7})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.6.31554496.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.6.924766416.2 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.34250608.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.855192924161485056.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 |
| $18$ | 18.6.74480276631114450998648208731279928449003802624.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.923825701716871234737684694141398098943.1 | \(\Z/9\Z\) | 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 | add | ss | ord | ss | ord | ord | ss | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | 2 | 1 | 1 | - | 1,1 | 1 | 1,1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 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.