Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-x-1\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-xz^2-z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108x-1026\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2, 1)$ | $0.91509546575046553472422421483$ | $\infty$ |
Integral points
\((2,\pm 1)\)
Invariants
| Conductor: | $N$ | = | \( 704 \) | = | $2^{6} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-704$ | = | $-1 \cdot 2^{6} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( -\frac{4096}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.76615520705548059728077786734$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1127287973354532519893939281$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8254556483942886$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.16671105127559$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.91509546575046553472422421483$ |
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| Real period: | $\Omega$ | ≈ | $2.0630782446896561300814265992$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.8879135472039337751259002355 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.887913547 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.063078 \cdot 0.915095 \cdot 1}{1^2} \\ & \approx 1.887913547\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 16 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | -1 | 6 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $5$ | 5B.4.1 | 25.60.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 316 & 545 \\ 377 & 287 \end{array}\right),\left(\begin{array}{rr} 291 & 30 \\ 85 & 11 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 191 & 539 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1699 & 2150 \\ 1700 & 2149 \end{array}\right),\left(\begin{array}{rr} 1099 & 0 \\ 0 & 2199 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2151 & 50 \\ 2150 & 51 \end{array}\right),\left(\begin{array}{rr} 1649 & 0 \\ 0 & 2199 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 1841 & 1639 \end{array}\right)$.
The torsion field $K:=\Q(E[2200])$ is a degree-$5068800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2200\Z)$.
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 | $2$ | \( 11 \) |
| $11$ | split multiplicative | $12$ | \( 64 = 2^{6} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 704k
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11a3, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/5\Z\) | 2.0.8.1-121.2-a3 |
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.247808.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.2.131153375232.4 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.7024111812608.1 | \(\Z/25\Z\) | not in database |
| $12$ | 12.2.1307761908383744.9 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.7430465388544.3 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.4.1505680748169532571648000000000000000.1 | \(\Z/5\Z\) | 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 | ord | ord | ord | split | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | 1 | 1 | 2 | 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.