Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+14184x+632560\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+14184xz^2+632560z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+226941x+40710782\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(8, 860\right) \) | $2.6732708478572310016249133942$ | $\infty$ |
| \( \left(-40, 20\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8:860:1]\) | $2.6732708478572310016249133942$ | $\infty$ |
| \([-40:20:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(31, 6912\right) \) | $2.6732708478572310016249133942$ | $\infty$ |
| \( \left(-161, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-40, 20\right) \), \( \left(8, 860\right) \), \( \left(8, -868\right) \)
\([-40:20:1]\), \([8:860:1]\), \([8:-868:1]\)
\( \left(-161, 0\right) \), \((31,\pm 6912)\)
Invariants
| Conductor: | $N$ | = | \( 9522 \) | = | $2 \cdot 3^{2} \cdot 23^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-357424956124272$ | = | $-1 \cdot 2^{4} \cdot 3^{8} \cdot 23^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{2924207}{3312} \) | = | $2^{-4} \cdot 3^{-2} \cdot 11^{3} \cdot 13^{3} \cdot 23^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4789203251158742316837679188$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.63813292718275545941723111557$ |
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| $abc$ quality: | $Q$ | ≈ | $0.898775602359137$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.398164910427257$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6732708478572310016249133942$ |
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| Real period: | $\Omega$ | ≈ | $0.35820347543457893381033300535$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.8302996339216145740312600386 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.830299634 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.358203 \cdot 2.673271 \cdot 16}{2^2} \\ & \approx 3.830299634\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 33792 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $23$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 |
|---|---|---|---|
| $2$ | 2B | 8.12.0.9 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 552 = 2^{3} \cdot 3 \cdot 23 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 512 & 549 \\ 435 & 182 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 307 & 486 \\ 258 & 163 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 546 & 547 \end{array}\right),\left(\begin{array}{rr} 73 & 72 \\ 174 & 79 \end{array}\right),\left(\begin{array}{rr} 367 & 0 \\ 0 & 551 \end{array}\right),\left(\begin{array}{rr} 545 & 8 \\ 544 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[552])$ is a degree-$410370048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/552\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 4761 = 3^{2} \cdot 23^{2} \) |
| $3$ | additive | $8$ | \( 1058 = 2 \cdot 23^{2} \) |
| $23$ | additive | $288$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 9522e
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 138c1, its twist by $69$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-23}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{69}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3069672194304.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.92844527616.6 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.49114755108864.55 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | ord | ss | ss | ord | ord | ord | add | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | 1 | 1,1 | 1,1 | 1 | 3 | 1 | - | 1 | 3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.