Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-30008x-1987488\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-30008xz^2-1987488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2430675x-1456170750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-98, 50)$ | $1.0934013665892114492821618169$ | $\infty$ |
| $(221, 1458)$ | $3.5731069323860288269177026990$ | $\infty$ |
| $(-103, 0)$ | $0$ | $2$ |
Integral points
\( \left(-103, 0\right) \), \((-98,\pm 50)\), \((221,\pm 1458)\), \((297,\pm 3900)\), \((302,\pm 4050)\), \((502,\pm 10450)\), \((4102,\pm 262450)\), \((2527502,\pm 4018252050)\)
Invariants
| Conductor: | $N$ | = | \( 13200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $5773680000000$ | = | $2^{10} \cdot 3^{8} \cdot 5^{7} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( \frac{186779563204}{360855} \) | = | $2^{2} \cdot 3^{-8} \cdot 5^{-1} \cdot 11^{-1} \cdot 13^{3} \cdot 277^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3376323741819838512459093266$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.044709232501687427235497107895$ |
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| $abc$ quality: | $Q$ | ≈ | $0.935175191781846$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.483707766575872$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.6707508141816088059247639263$ |
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| Real period: | $\Omega$ | ≈ | $0.36277678736339652874197244635$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $5.3266527503215047234186327360 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.326652750 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.362777 \cdot 3.670751 \cdot 16}{2^2} \\ & \approx 5.326652750\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49152 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 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 | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 440 = 2^{3} \cdot 5 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 368 & 3 \\ 285 & 2 \end{array}\right),\left(\begin{array}{rr} 433 & 8 \\ 432 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 172 & 439 \\ 329 & 434 \end{array}\right),\left(\begin{array}{rr} 384 & 157 \\ 381 & 352 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 434 & 435 \end{array}\right),\left(\begin{array}{rr} 59 & 58 \\ 178 & 395 \end{array}\right)$.
The torsion field $K:=\Q(E[440])$ is a degree-$202752000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/440\Z)$.
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 | $2$ | \( 275 = 5^{2} \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 13200.b
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1320.e1, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{55}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | 4.2.2662000.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{55})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7086244000000.2 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4535196160000.19 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.123904000000.9 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | nonsplit | add | ord | split | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 4 | - | 4 | 3 | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2,4 |
| $\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.