Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-21826x+766548\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-21826xz^2+766548z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-28285875x+35848932750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-27, 1169)$ | $0.56016342200946083449162639666$ | $\infty$ |
| $(-163, 81)$ | $0$ | $2$ |
Integral points
\( \left(-163, 81\right) \), \( \left(-27, 1169\right) \), \( \left(-27, -1143\right) \), \( \left(37, 81\right) \), \( \left(37, -119\right) \), \( \left(126, 81\right) \), \( \left(126, -208\right) \), \( \left(262, 3481\right) \), \( \left(262, -3744\right) \), \( \left(7758, 679329\right) \), \( \left(7758, -687088\right) \)
Invariants
| Conductor: | $N$ | = | \( 14450 \) | = | $2 \cdot 5^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $410338673000000$ | = | $2^{6} \cdot 5^{6} \cdot 17^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{3048625}{1088} \) | = | $2^{-6} \cdot 5^{3} \cdot 17^{-1} \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5049332916966962407731587976$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.71639233654846198665198817795$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9000957170016773$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.341632528930862$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.56016342200946083449162639666$ |
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| Real period: | $\Omega$ | ≈ | $0.48762315178788718324033370314$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.0925946134261664985887871699 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.092594613 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.487623 \cdot 0.560163 \cdot 16}{2^2} \\ & \approx 1.092594613\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 82944 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $4$ | $I_{1}^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1021 & 420 \\ 1230 & 481 \end{array}\right),\left(\begin{array}{rr} 2029 & 12 \\ 2028 & 13 \end{array}\right),\left(\begin{array}{rr} 281 & 410 \\ 1350 & 421 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1990 & 2031 \end{array}\right),\left(\begin{array}{rr} 854 & 405 \\ 795 & 824 \end{array}\right),\left(\begin{array}{rr} 1223 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1251 & 1720 \\ 2030 & 1301 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$28877783040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 | $4$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $3$ | good | $2$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 578 = 2 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 14450g
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 34a1, its twist by $85$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{85}) \) | \(\Z/6\Z\) | 2.2.85.1-68.1-b1 |
| $4$ | 4.0.27200.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.4792017375.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.213813760000.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.15448044160000.12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.213813760000.35 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | 16.0.45716323965337600000000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.30697891718776609628345544000000000.1 | \(\Z/18\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | ord | add | ord | ord | ord | add | ord | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 8 | 3 | - | 1 | 1 | 1 | - | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0 | 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.