Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-5106438x+873211492\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-5106438xz^2+873211492z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6617943675x+40760409201750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(172, -86)$ | $0$ | $2$ |
Integral points
\( \left(172, -86\right) \)
Invariants
| Conductor: | $N$ | = | \( 41650 \) | = | $2 \cdot 5^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $8192135168000000000000$ | = | $2^{24} \cdot 5^{12} \cdot 7^{6} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{8010684753304969}{4456448000000} \) | = | $2^{-24} \cdot 5^{-6} \cdot 17^{-1} \cdot 19^{3} \cdot 10531^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8951505077867400352181162552$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1174764770420331953650602169$ |
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| $abc$ quality: | $Q$ | ≈ | $1.042560452096738$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.448089911383551$ | |||
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: | $\Omega$ | ≈ | $0.11352228552096106010991148165$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ ( 2^{3} \cdot 3 )\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.7245348525030654426378755596 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.724534853 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.113522 \cdot 1.000000 \cdot 96}{2^2} \\ & \approx 2.724534853\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4147200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $5$ | $2$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $1$ | $I_{1}$ | 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 | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 6378 & 4081 \\ 14063 & 8 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 14265 & 10094 \\ 6202 & 11269 \end{array}\right),\left(\begin{array}{rr} 7141 & 12264 \\ 13272 & 4369 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 8159 & 0 \\ 0 & 14279 \end{array}\right),\left(\begin{array}{rr} 9521 & 12264 \\ 9520 & 1 \end{array}\right),\left(\begin{array}{rr} 10711 & 12264 \\ 6132 & 4369 \end{array}\right),\left(\begin{array}{rr} 14257 & 24 \\ 14256 & 25 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 14180 & 14261 \end{array}\right)$.
The torsion field $K:=\Q(E[14280])$ is a degree-$14554402652160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14280\Z)$.
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 | $4$ | \( 20825 = 5^{2} \cdot 7^{2} \cdot 17 \) |
| $3$ | good | $2$ | \( 20825 = 5^{2} \cdot 7^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $26$ | \( 850 = 2 \cdot 5^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 41650.bp
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 170.a1, its twist by $-35$.
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 |
| $2$ | \(\Q(\sqrt{105}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.333200.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.32228665875.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.579543031690000.4 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.32085427360000.26 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.8992801440000.46 | \(\Z/12\Z\) | 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 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.268322827408063581050058559883625000000000000.1 | \(\Z/18\Z\) | 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 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | ord | add | add | split |
| $\lambda$-invariant(s) | 6 | 8 | - | - | 1 |
| $\mu$-invariant(s) | 0 | 0 | - | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ 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 $0$.