Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x+137\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-xz^2+137z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1323x+6395814\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{7}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2, 11\right) \) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([2:11:1]\) | $0$ | $7$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(75, 2592\right) \) | $0$ | $7$ |
Integral points
\( \left(-4, 11\right) \), \( \left(-4, -7\right) \), \( \left(2, 11\right) \), \( \left(2, -13\right) \), \( \left(14, 47\right) \), \( \left(14, -61\right) \)
\([-4:11:1]\), \([-4:-7:1]\), \([2:11:1]\), \([2:-13:1]\), \([14:47:1]\), \([14:-61:1]\)
\((-141,\pm 1944)\), \((75,\pm 2592)\), \((507,\pm 11664)\)
Invariants
| Conductor: | $N$ | = | \( 174 \) | = | $2 \cdot 3 \cdot 29$ |
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| Minimal Discriminant: | $\Delta$ | = | $-8118144$ | = | $-1 \cdot 2^{7} \cdot 3^{7} \cdot 29 $ |
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| j-invariant: | $j$ | = | \( -\frac{117649}{8118144} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 7^{6} \cdot 29^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.0046914138593795064534520490529$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.0046914138593795064534520490529$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2436350520132442$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.528802095628653$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $1.8598039585135014719853739017$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 49 $ = $ 7\cdot7\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $7$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.8598039585135014719853739017 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.859803959 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.859804 \cdot 1.000000 \cdot 49}{7^2} \\ & \approx 1.859803959\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 28 |
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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 semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $29$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $7$ | 7B.1.1 | 7.48.0.1 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4872 = 2^{3} \cdot 3 \cdot 7 \cdot 29 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 4033 & 14 \\ 3871 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2437 & 14 \\ 2443 & 99 \end{array}\right),\left(\begin{array}{rr} 1219 & 2450 \\ 0 & 3307 \end{array}\right),\left(\begin{array}{rr} 3655 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 1625 & 14 \\ 1631 & 99 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 4859 & 14 \\ 4858 & 15 \end{array}\right)$.
The torsion field $K:=\Q(E[4872])$ is a degree-$1056056279040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4872\Z)$.
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 | $4$ | \( 87 = 3 \cdot 29 \) |
| $3$ | split multiplicative | $4$ | \( 58 = 2 \cdot 29 \) |
| $7$ | good | $2$ | \( 29 \) |
| $29$ | split multiplicative | $30$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 174b
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{7}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.696.1 | \(\Z/14\Z\) | not in database |
| $6$ | 6.0.337153536.1 | \(\Z/2\Z \oplus \Z/14\Z\) | not in database |
| $8$ | 8.2.2004683316912.3 | \(\Z/21\Z\) | not in database |
| $12$ | deg 12 | \(\Z/28\Z\) | 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 | 29 |
|---|---|---|---|---|---|
| Reduction type | split | split | ord | ord | split |
| $\lambda$-invariant(s) | 6 | 1 | 0 | 2 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.