Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-159x+665\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-159xz^2+665z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2547x+40014\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1, 22\right) \) | $0.22625814982945270105876354109$ | $\infty$ |
| \( \left(-14, 7\right) \) | $0$ | $2$ |
| \( \left(10, -5\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1:22:1]\) | $0.22625814982945270105876354109$ | $\infty$ |
| \([-14:7:1]\) | $0$ | $2$ |
| \([10:-5:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3, 180\right) \) | $0.22625814982945270105876354109$ | $\infty$ |
| \( \left(-57, 0\right) \) | $0$ | $2$ |
| \( \left(39, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-14, 7\right) \), \( \left(-11, 37\right) \), \( \left(-11, -26\right) \), \( \left(-4, 37\right) \), \( \left(-4, -33\right) \), \( \left(1, 22\right) \), \( \left(1, -23\right) \), \( \left(4, 7\right) \), \( \left(4, -11\right) \), \( \left(10, -5\right) \), \( \left(11, 7\right) \), \( \left(11, -18\right) \), \( \left(16, 37\right) \), \( \left(16, -53\right) \), \( \left(31, 142\right) \), \( \left(31, -173\right) \), \( \left(35, 175\right) \), \( \left(35, -210\right) \), \( \left(106, 1027\right) \), \( \left(106, -1133\right) \), \( \left(136, 1507\right) \), \( \left(136, -1643\right) \), \( \left(7591, 657547\right) \), \( \left(7591, -665138\right) \)
\([-14:7:1]\), \([-11:37:1]\), \([-11:-26:1]\), \([-4:37:1]\), \([-4:-33:1]\), \([1:22:1]\), \([1:-23:1]\), \([4:7:1]\), \([4:-11:1]\), \([10:-5:1]\), \([11:7:1]\), \([11:-18:1]\), \([16:37:1]\), \([16:-53:1]\), \([31:142:1]\), \([31:-173:1]\), \([35:175:1]\), \([35:-210:1]\), \([106:1027:1]\), \([106:-1133:1]\), \([136:1507:1]\), \([136:-1643:1]\), \([7591:657547:1]\), \([7591:-665138:1]\)
\( \left(-57, 0\right) \), \((-45,\pm 252)\), \((-17,\pm 280)\), \((3,\pm 180)\), \((15,\pm 72)\), \( \left(39, 0\right) \), \((43,\pm 100)\), \((63,\pm 360)\), \((123,\pm 1260)\), \((139,\pm 1540)\), \((423,\pm 8640)\), \((543,\pm 12600)\), \((30363,\pm 5290740)\)
Invariants
| Conductor: | $N$ | = | \( 630 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $89302500$ | = | $2^{2} \cdot 3^{6} \cdot 5^{4} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{611960049}{122500} \) | = | $2^{-2} \cdot 3^{3} \cdot 5^{-4} \cdot 7^{-2} \cdot 283^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.24172022566871843738671254202$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.30758591866533640831091007644$ |
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| $abc$ quality: | $Q$ | ≈ | $1.02632404766684$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.16149817903497$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.22625814982945270105876354109$ |
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| Real period: | $\Omega$ | ≈ | $1.8098438078040272002312179947$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.6379676457361231357899833054 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.637967646 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.809844 \cdot 0.226258 \cdot 64}{4^2} \\ & \approx 1.637967646\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 256 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 8.12.0.3 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 163 & 114 \\ 78 & 55 \end{array}\right),\left(\begin{array}{rr} 55 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 165 & 4 \\ 164 & 5 \end{array}\right),\left(\begin{array}{rr} 85 & 114 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 114 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$3096576$ 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 | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $6$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 630e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 70a2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{3}, \sqrt{-7})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.52509870000.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | \(\Q(i, \sqrt{2}, \sqrt{3}, \sqrt{7})\) | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | split | nonsplit | ord | ord | ord | ss | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 2 | 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 | 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.