Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+27x+27\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+27xz^2+27z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2160x+13176\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3, 12)$ | $0.58411882142680819305671450004$ | $\infty$ |
| $(-1, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1, 0\right) \), \((3,\pm 12)\), \((6,\pm 21)\), \((111,\pm 1176)\)
Invariants
| Conductor: | $N$ | = | \( 1344 \) | = | $2^{6} \cdot 3 \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-1354752$ | = | $-1 \cdot 2^{10} \cdot 3^{3} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2048000}{1323} \) | = | $2^{14} \cdot 3^{-3} \cdot 5^{3} \cdot 7^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.13449718188272894210785368001$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.71211983234935003328888044789$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1084282779180015$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.979680375888891$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.58411882142680819305671450004$ |
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| Real period: | $\Omega$ | ≈ | $1.6898990654296101057556275696$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9613055512870254400837147860 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.961305551 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.689899 \cdot 0.584119 \cdot 12}{2^2} \\ & \approx 2.961305551\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $2$ | $I_{2}$ | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 157 & 12 \\ 156 & 13 \end{array}\right),\left(\begin{array}{rr} 157 & 166 \\ 134 & 93 \end{array}\right),\left(\begin{array}{rr} 95 & 156 \\ 96 & 155 \end{array}\right),\left(\begin{array}{rr} 83 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 73 & 12 \\ 18 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 39 & 10 \\ 14 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 118 & 159 \end{array}\right),\left(\begin{array}{rr} 62 & 11 \\ 45 & 64 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$1548288$ 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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 3 \) |
| $3$ | split multiplicative | $4$ | \( 448 = 2^{6} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 1344.o
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 84.b4, its twist by $-8$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/6\Z\) | 2.0.8.1-1764.2-b2 |
| $4$ | 4.2.37632.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.132765696.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.2341011456.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.12745506816.11 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5664669696.1 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.2599167103947239325696.3 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.10195577227840027058831892676608.2 | \(\Z/18\Z\) | 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 | add | split | ss | nonsplit | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 1,1 | 3 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.