Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-206988x-32631408\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-206988xz^2-32631408z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-268256475x-1521646202250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-294, 1830)$ | $0.15977890749593835707824220560$ | $\infty$ |
| $(-328, 164)$ | $0$ | $2$ |
Integral points
\( \left(-328, 164\right) \), \( \left(-294, 1830\right) \), \( \left(-294, -1536\right) \), \( \left(-228, 1764\right) \), \( \left(-228, -1536\right) \), \( \left(-192, 300\right) \), \( \left(-192, -108\right) \), \( \left(522, 1014\right) \), \( \left(522, -1536\right) \), \( \left(696, 12324\right) \), \( \left(696, -13020\right) \), \( \left(828, 18660\right) \), \( \left(828, -19488\right) \), \( \left(3072, 166764\right) \), \( \left(3072, -169836\right) \), \( \left(56316, 13335780\right) \), \( \left(56316, -13392096\right) \)
Invariants
| Conductor: | $N$ | = | \( 28050 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $108053790372000000$ | = | $2^{8} \cdot 3^{5} \cdot 5^{6} \cdot 11^{3} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{62768149033310713}{6915442583808} \) | = | $2^{-8} \cdot 3^{-5} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 37^{3} \cdot 467^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0018377647977946977891907013$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1971188085807445104888110347$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9879715979533532$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.719396623620909$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.15977890749593835707824220560$ |
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| Real period: | $\Omega$ | ≈ | $0.22542826790314827617743593636$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 960 $ = $ 2^{3}\cdot5\cdot2\cdot3\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.6444837674240171798670515381 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.644483767 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.225428 \cdot 0.159779 \cdot 960}{2^2} \\ & \approx 8.644483767\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 491520 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20761 & 20760 \\ 9550 & 11791 \end{array}\right),\left(\begin{array}{rr} 16736 & 17955 \\ 2045 & 13466 \end{array}\right),\left(\begin{array}{rr} 13463 & 0 \\ 0 & 22439 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 14521 & 17960 \\ 22180 & 4521 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 22434 & 22435 \end{array}\right),\left(\begin{array}{rr} 2996 & 13465 \\ 10495 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 22433 & 8 \\ 22432 & 9 \end{array}\right),\left(\begin{array}{rr} 3931 & 3930 \\ 20770 & 571 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$762373472256000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
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 | $4$ | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 850 = 2 \cdot 5^{2} \cdot 17 \) |
| $5$ | additive | $14$ | \( 374 = 2 \cdot 11 \cdot 17 \) |
| $11$ | split multiplicative | $12$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 28050.cy
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1122.c3, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.206634875040000.30 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4604564930625.5 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.147954945870000.9 | \(\Z/6\Z\) | 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 | split | split | add | ord | split | ord | split | ord | ss | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | 2 | - | 1 | 2 | 1 | 2 | 3 | 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.