Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-849x-6831\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-849xz^2-6831z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-68796x-5186160\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-16, 49\right) \) | $1.3182432765968104593130687615$ | $\infty$ |
| \( \left(-21, 36\right) \) | $2.0709966913819543123166491522$ | $\infty$ |
| \( \left(-9, 0\right) \) | $0$ | $2$ |
| \( \left(33, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-16:49:1]\) | $1.3182432765968104593130687615$ | $\infty$ |
| \([-21:36:1]\) | $2.0709966913819543123166491522$ | $\infty$ |
| \([-9:0:1]\) | $0$ | $2$ |
| \([33:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-147, 1323\right) \) | $1.3182432765968104593130687615$ | $\infty$ |
| \( \left(-192, 972\right) \) | $2.0709966913819543123166491522$ | $\infty$ |
| \( \left(-84, 0\right) \) | $0$ | $2$ |
| \( \left(294, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-23, 0\right) \), \((-21,\pm 36)\), \((-16,\pm 49)\), \((-15,\pm 48)\), \( \left(-9, 0\right) \), \( \left(33, 0\right) \), \((40,\pm 147)\), \((41,\pm 160)\), \((75,\pm 588)\), \((89,\pm 784)\), \((233,\pm 3520)\), \((327,\pm 5880)\), \((369,\pm 7056)\), \((9441,\pm 917280)\), \((94169,\pm 28897456)\)
\([-23:0:1]\), \([-21:\pm 36:1]\), \([-16:\pm 49:1]\), \([-15:\pm 48:1]\), \([-9:0:1]\), \([33:0:1]\), \([40:\pm 147:1]\), \([41:\pm 160:1]\), \([75:\pm 588:1]\), \([89:\pm 784:1]\), \([233:\pm 3520:1]\), \([327:\pm 5880:1]\), \([369:\pm 7056:1]\), \([9441:\pm 917280:1]\), \([94169:\pm 28897456:1]\)
\( \left(-23, 0\right) \), \((-21,\pm 36)\), \((-16,\pm 49)\), \((-15,\pm 48)\), \( \left(-9, 0\right) \), \( \left(33, 0\right) \), \((40,\pm 147)\), \((41,\pm 160)\), \((75,\pm 588)\), \((89,\pm 784)\), \((233,\pm 3520)\), \((327,\pm 5880)\), \((369,\pm 7056)\), \((9441,\pm 917280)\), \((94169,\pm 28897456)\)
Invariants
| Conductor: | $N$ | = | \( 9408 \) | = | $2^{6} \cdot 3 \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $17348050944$ | = | $2^{14} \cdot 3^{2} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{35152}{9} \) | = | $2^{4} \cdot 3^{-2} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.67417637872777203460228175562$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1074504064531541456038320911$ |
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| $abc$ quality: | $Q$ | ≈ | $0.972547111469975$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4808022113323513$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5929144509875476566860852402$ |
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| Real period: | $\Omega$ | ≈ | $0.90107146676304894214219549177$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.6728024550849106872925107386 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.672802455 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.901071 \cdot 2.592914 \cdot 32}{4^2} \\ & \approx 4.672802455\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6144 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}^{*}$ | additive | 1 | 6 | 14 | 0 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 8.48.0.138 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 71 & 98 \\ 126 & 43 \end{array}\right),\left(\begin{array}{rr} 55 & 0 \\ 56 & 139 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 164 & 165 \end{array}\right),\left(\begin{array}{rr} 95 & 0 \\ 0 & 167 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 161 & 8 \\ 160 & 9 \end{array}\right),\left(\begin{array}{rr} 29 & 56 \\ 70 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$774144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 | $2$ | \( 49 = 7^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3136 = 2^{6} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 192 = 2^{6} \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 9408q
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 24a1, its twist by $-56$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.56.1-72.2-a3 |
| $4$ | \(\Q(\sqrt{3}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.12745506816.9 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.458838245376.7 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.157351936.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.12745506816.6 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.27874423406592.6 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.162447943996702457856.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | nonsplit | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 4 | 2 | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
| $\mu$-invariant(s) | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.