Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-8x-16\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-8xz^2-16z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-675x-13662\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 0)$ | $0$ | $2$ |
Integral points
\( \left(4, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 112 \) | = | $2^{4} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-114688$ | = | $-1 \cdot 2^{14} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( -\frac{15625}{28} \) | = | $-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.33893771796505257910584682136$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0320848985249978885230789428$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0171207213859017$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.108201712646352$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $1.3254912396824867143349662920$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3254912396824867143349662920 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.325491240 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.325491 \cdot 1.000000 \cdot 4}{2^2} \\ & \approx 1.325491240\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 8 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 4 | 14 | 2 |
| $7$ | $1$ | $I_{1}$ | nonsplit 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
| $3$ | 3B | 9.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 253 & 36 \\ 10 & 361 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 281 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 216 & 91 \end{array}\right),\left(\begin{array}{rr} 377 & 468 \\ 242 & 143 \end{array}\right),\left(\begin{array}{rr} 160 & 9 \\ 47 & 22 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$13934592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 | $2$ | \( 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 16 = 2^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 112c
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a4, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.7.1-1792.5-b6 |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/6\Z\) | 2.0.4.1-98.1-a2 |
| $4$ | \(\Q(i, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.2.1792.1 | \(\Z/4\Z\) | not in database |
| $6$ | 6.2.16595712.3 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.153664.1 | \(\Z/18\Z\) | not in database |
| $8$ | 8.0.157351936.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.30118144.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3211264.1 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.275417656786944.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.114709561344.1 | \(\Z/18\Z\) | not in database |
| $12$ | 12.0.13495465182560256.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | \(\Q(\zeta_{28})\) | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | 16.0.24759631762948096.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.232218265089212416.1 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 |
|---|---|---|---|
| Reduction type | add | ord | nonsplit |
| $\lambda$-invariant(s) | - | 0 | 0 |
| $\mu$-invariant(s) | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.