Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+39x-19\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+39xz^2-19z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+621x-594\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10, 31\right) \) | $0.10197829462292110456679008894$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([10:31:1]\) | $0.10197829462292110456679008894$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(39, 288\right) \) | $0.10197829462292110456679008894$ | $\infty$ |
Integral points
\( \left(1, 4\right) \), \( \left(1, -5\right) \), \( \left(2, 7\right) \), \( \left(2, -9\right) \), \( \left(10, 31\right) \), \( \left(10, -41\right) \), \( \left(19, 76\right) \), \( \left(19, -95\right) \), \( \left(154, 1831\right) \), \( \left(154, -1985\right) \)
\([1:4:1]\), \([1:-5:1]\), \([2:7:1]\), \([2:-9:1]\), \([10:31:1]\), \([10:-41:1]\), \([19:76:1]\), \([19:-95:1]\), \([154:1831:1]\), \([154:-1985:1]\)
\((3,\pm 36)\), \((7,\pm 64)\), \((39,\pm 288)\), \((75,\pm 684)\), \((615,\pm 15264)\)
Invariants
| Conductor: | $N$ | = | \( 162 \) | = | $2 \cdot 3^{4}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-3779136$ | = | $-1 \cdot 2^{6} \cdot 3^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{109503}{64} \) | = | $2^{-6} \cdot 3^{2} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.048500341608534496996168113703$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.96401058216529257315887247781$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2854862760382055$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.440177369294559$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.10197829462292110456679008894$ |
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| Real period: | $\Omega$ | ≈ | $1.4651475876169562933434955407$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $0.89647951413638449139673303664 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 0.896479514 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.465148 \cdot 0.101978 \cdot 6}{1^2} \\ & \approx 0.896479514\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 36 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $3$ | $IV^{*}$ | additive | 1 | 4 | 10 | 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 |
|---|---|---|---|
| $2$ | 2G | 4.8.0.2 | $8$ |
| $3$ | 3B.1.2 | 3.8.0.2 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 12.128.1-12.b.1.3, level \( 12 = 2^{2} \cdot 3 \), index $128$, genus $1$, and generators
$\left(\begin{array}{rr} 9 & 4 \\ 8 & 5 \end{array}\right),\left(\begin{array}{rr} 7 & 3 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 8 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[12])$ is a degree-$36$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12\Z)$.
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 | $4$ | \( 81 = 3^{4} \) |
| $3$ | additive | $4$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 162.a
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-2916.1-a2 |
| $3$ | 3.1.324.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.243.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.419904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1259712.2 | \(\Z/4\Z\) | not in database |
| $6$ | 6.0.314928.2 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.1.74384733888.1 | \(\Z/6\Z\) | not in database |
| $12$ | 12.0.1586874322944.4 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.16599265906765726789632.7 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.16599265906765726789632.5 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.354117672677668838178816.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.1062353018033006514536448.1 | \(\Z/12\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | add | ord | ord | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | 1 | 1 | 1,1 | 3 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-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.