Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-7151600x-21211722015\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-7151600xz^2-21211722015z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-9268473627x-989515075219146\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(34783, 6448983)$ | $4.7439573448504413430169667497$ | $\infty$ |
Integral points
\( \left(34783, 6448983\right) \), \( \left(34783, -6483767\right) \)
Invariants
| Conductor: | $N$ | = | \( 141610 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-170908877254445524250000$ | = | $-1 \cdot 2^{4} \cdot 5^{6} \cdot 7^{8} \cdot 17^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{290707016929}{1228250000} \) | = | $-1 \cdot 2^{-4} \cdot 5^{-6} \cdot 7 \cdot 17^{-3} \cdot 3463^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1430533333245640249590585985$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.42917322859291378143072279393$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9321837580203192$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.149596902731847$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.7439573448504413430169667497$ |
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| Real period: | $\Omega$ | ≈ | $0.042013715779889472271137283887$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot( 2 \cdot 3 )\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.5669412268063464173725636089 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.566941227 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.042014 \cdot 4.743957 \cdot 48}{1^2} \\ & \approx 9.566941227\end{aligned}$$
Modular invariants
Modular form 141610.2.a.cc
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13934592 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $1$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
| $17$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 204 = 2^{2} \cdot 3 \cdot 17 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 103 & 6 \\ 105 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 199 & 6 \\ 198 & 7 \end{array}\right),\left(\begin{array}{rr} 201 & 202 \\ 194 & 197 \end{array}\right),\left(\begin{array}{rr} 167 & 198 \\ 93 & 185 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 122 & 87 \\ 163 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[204])$ is a degree-$22560768$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/204\Z)$.
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 | $4$ | \( 14161 = 7^{2} \cdot 17^{2} \) |
| $3$ | good | $2$ | \( 28322 = 2 \cdot 7^{2} \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 28322 = 2 \cdot 7^{2} \cdot 17^{2} \) |
| $7$ | additive | $26$ | \( 2890 = 2 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 141610.cc
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 8330.r1, its twist by $-119$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-51}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.3332.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.754951232.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1698640272.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.5095920816.8 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.18882704448220931948132089389991818453430687500000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.1254712469737791050282343333888.1 | \(\Z/6\Z\) | 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 | ord | split | add | ord | ord | add | ord | ss | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 7 | 1 | 2 | - | 1 | 1 | - | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 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.