Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-3388040x-2355457260\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-3388040xz^2-2355457260z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-274431267x-1716305048766\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1181, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1181, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 29040 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $94489056709419018240$ | = | $2^{11} \cdot 3^{16} \cdot 5 \cdot 11^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{1185450336504002}{26043266205} \) | = | $2 \cdot 3^{-16} \cdot 5^{-1} \cdot 11^{-2} \cdot 167^{3} \cdot 503^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6209093574552821536652715983$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.78657680554281368133517036465$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9967634929262502$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.519513360404227$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.11142765903868978033279878507$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2^{2}\cdot2^{4}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.5656850892380729706495611222 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.565685089 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.111428 \cdot 1.000000 \cdot 128}{2^2} \\ & \approx 3.565685089\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 983040 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
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 | 16.48.0.113 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 880 = 2^{4} \cdot 5 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 192 & 5 \\ 835 & 866 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 782 & 867 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 676 & 661 \\ 599 & 450 \end{array}\right),\left(\begin{array}{rr} 109 & 864 \\ 330 & 219 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 876 & 877 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 227 & 864 \\ 336 & 315 \end{array}\right),\left(\begin{array}{rr} 865 & 16 \\ 864 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[880])$ is a degree-$811008000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/880\Z)$.
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 | $4$ | \( 605 = 5 \cdot 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \) |
| $5$ | split multiplicative | $6$ | \( 5808 = 2^{4} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 29040.dg
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 1320.f2, its twist by $44$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-110}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{-11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-11})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-11})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.3838050304000000.16 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3838050304000000.22 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.599695360000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/16\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | 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 | 11 |
|---|---|---|---|---|
| Reduction type | add | split | split | add |
| $\lambda$-invariant(s) | - | 11 | 1 | - |
| $\mu$-invariant(s) | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.