Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-43x+105\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-43xz^2+105z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-55755x+5066118\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8, 13)$ | $0$ | $6$ |
Integral points
\( \left(4, -1\right) \), \( \left(4, -3\right) \), \( \left(8, 13\right) \), \( \left(8, -21\right) \)
Invariants
| Conductor: | $N$ | = | \( 34 \) | = | $2 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $2312$ | = | $2^{3} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{8805624625}{2312} \) | = | $2^{-3} \cdot 5^{3} \cdot 7^{3} \cdot 17^{-2} \cdot 59^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.36981874626848933194337211720$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.36981874626848933194337211720$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9659043939476677$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.493566484861295$ | |||
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: | $\Omega$ | ≈ | $4.4956633263137035532067468518$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.74927722105228392553445780863 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.749277221 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 4.495663 \cdot 1.000000 \cdot 6}{6^2} \\ & \approx 0.749277221\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4 |
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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 semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $2$ | $I_{2}$ | 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 | 8.6.0.6 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 408 = 2^{3} \cdot 3 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 10 & 3 \\ 177 & 400 \end{array}\right),\left(\begin{array}{rr} 27 & 88 \\ 398 & 77 \end{array}\right),\left(\begin{array}{rr} 241 & 12 \\ 222 & 73 \end{array}\right),\left(\begin{array}{rr} 397 & 12 \\ 396 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 358 & 399 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 281 & 2 \\ 126 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[408])$ is a degree-$60162048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/408\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 1 \) |
| $3$ | good | $2$ | \( 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 34a
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.2.8.1-578.1-d6 |
| $4$ | 4.4.9248.1 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.1212153856.6 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.8.5473632256.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.105212405952.1 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.23214739404794772903803486208.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | 17 |
|---|---|---|---|
| Reduction type | split | ord | nonsplit |
| $\lambda$-invariant(s) | 1 | 2 | 0 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.