Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-87952x-9822928\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-87952xz^2-9822928z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-7124139x-7139542122\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(419, 5202\right) \) | $1.8983988180274677484957193068$ | $\infty$ |
| \( \left(-193, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([419:5202:1]\) | $1.8983988180274677484957193068$ | $\infty$ |
| \([-193:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3774, 140454\right) \) | $1.8983988180274677484957193068$ | $\infty$ |
| \( \left(-1734, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-193, 0\right) \), \((419,\pm 5202)\)
\([-193:0:1]\), \([419:\pm 5202:1]\)
\( \left(-193, 0\right) \), \((419,\pm 5202)\)
Invariants
| Conductor: | $N$ | = | \( 90168 \) | = | $2^{3} \cdot 3 \cdot 13 \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $2108176048862208$ | = | $2^{10} \cdot 3^{8} \cdot 13 \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3044193988}{85293} \) | = | $2^{2} \cdot 3^{-8} \cdot 11^{3} \cdot 13^{-1} \cdot 83^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7194401372839665275238699369$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.27478918521076260378192413992$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9567914796065754$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.011353272736901$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.8983988180274677484957193068$ |
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| Real period: | $\Omega$ | ≈ | $0.27770334358848219999691784099$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 64 $ = $ 2\cdot2^{3}\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.4350671876904059597807549878 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.435067188 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.277703 \cdot 1.898399 \cdot 64}{2^2} \\ & \approx 8.435067188\end{aligned}$$
Modular invariants
Modular form 90168.2.a.bc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 655360 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|---|
| $2$ | 2B | 8.12.0.10 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10608 = 2^{4} \cdot 3 \cdot 13 \cdot 17 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 10593 & 16 \\ 10592 & 17 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10303 & 6562 \\ 7786 & 4727 \end{array}\right),\left(\begin{array}{rr} 3656 & 7497 \\ 5831 & 3146 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 7073 & 6256 \\ 6664 & 7617 \end{array}\right),\left(\begin{array}{rr} 7957 & 7650 \\ 10064 & 6019 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 10322 & 7443 \end{array}\right),\left(\begin{array}{rr} 8735 & 0 \\ 0 & 10607 \end{array}\right)$.
The torsion field $K:=\Q(E[10608])$ is a degree-$12613815631872$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10608\Z)$.
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 | $2$ | \( 3757 = 13 \cdot 17^{2} \) |
| $3$ | split multiplicative | $4$ | \( 30056 = 2^{3} \cdot 13 \cdot 17^{2} \) |
| $13$ | split multiplicative | $14$ | \( 6936 = 2^{3} \cdot 3 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 90168o
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 312d3, its twist by $17$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-221}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{-17})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.1651261089746944.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.57815240704.22 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.26420177435951104.54 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | split | ord | ord | ss | split | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | 1 | 1 | 1,3 | 2 | - | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-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.