Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-32693x-2267130\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-32693xz^2-2267130z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-523083x-145619386\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(212, 438)$ | $1.4603861251154790483628228560$ | $\infty$ |
Integral points
\( \left(212, 438\right) \), \( \left(212, -651\right) \), \( \left(8319, 754389\right) \), \( \left(8319, -762709\right) \)
Invariants
| Conductor: | $N$ | = | \( 1089 \) | = | $3^{2} \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-156267624249$ | = | $-1 \cdot 3^{6} \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( -24729001 \) | = | $-1 \cdot 11 \cdot 131^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2302929426175610283436549496$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.91761005024874084672859672084$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9685335876741098$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.120163088375376$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4603861251154790483628228560$ |
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| Real period: | $\Omega$ | ≈ | $0.17752349909483497806512248401$ |
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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)$ | ≈ | $1.5555171297602838352109652600 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.555517130 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.177523 \cdot 1.460386 \cdot 6}{1^2} \\ & \approx 1.555517130\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1584 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $11$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 |
|---|---|---|
| $2$ | 2G | 4.2.0.1 |
| $11$ | 11B.10.5 | 11.60.1.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $480$, genus $16$, and generators
$\left(\begin{array}{rr} 177 & 88 \\ 176 & 177 \end{array}\right),\left(\begin{array}{rr} 175 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 133 & 132 \\ 66 & 1 \end{array}\right),\left(\begin{array}{rr} 133 & 33 \\ 0 & 67 \end{array}\right),\left(\begin{array}{rr} 1 & 108 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 71 & 132 \\ 198 & 95 \end{array}\right),\left(\begin{array}{rr} 133 & 132 \\ 132 & 133 \end{array}\right),\left(\begin{array}{rr} 1 & 132 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 88 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 132 & 1 \end{array}\right),\left(\begin{array}{rr} 100 & 231 \\ 165 & 199 \end{array}\right),\left(\begin{array}{rr} 133 & 198 \\ 198 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$2027520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $6$ | \( 121 = 11^{2} \) |
| $11$ | additive | $52$ | \( 9 = 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 1089h
consists of 2 curves linked by isogenies of
degree 11.
Twists
The minimal quadratic twist of this elliptic curve is 121a1, its twist by $33$.
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 |
|---|---|---|---|
| $3$ | 3.1.484.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.937024.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.32019867.2 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.52089208083.1 | \(\Z/11\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | ord | add | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | ? | - | 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 |
An entry ? indicates that the invariants have not yet been computed.
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.