Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-5048733x+4364278259\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-5048733xz^2+4364278259z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-6543157995x+203717913830118\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1281, 346\right) \) | $3.0552475265220546043050847513$ | $\infty$ |
| \( \left(\frac{5187}{4}, -\frac{5191}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1281:346:1]\) | $3.0552475265220546043050847513$ | $\infty$ |
| \([10374:-5191:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(46131, 213192\right) \) | $3.0552475265220546043050847513$ | $\infty$ |
| \( \left(46698, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(1281, 346\right) \), \( \left(1281, -1628\right) \)
\([1281:346:1]\), \([1281:-1628:1]\)
\((46131,\pm 213192)\)
Invariants
| Conductor: | $N$ | = | \( 25886 \) | = | $2 \cdot 7 \cdot 43^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $158590356173312$ | = | $2^{9} \cdot 7^{2} \cdot 43^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{2251439055699625}{25088} \) | = | $2^{-9} \cdot 5^{3} \cdot 7^{-2} \cdot 11^{3} \cdot 2383^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2937010382698656693172036115$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.41310098042308445758078235483$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0648884149541868$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.699728246017661$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0552475265220546043050847513$ |
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| Real period: | $\Omega$ | ≈ | $0.40427103807994209896552581636$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 36 $ = $ 3^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.116332802244218004140499385 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.116332802 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.404271 \cdot 3.055248 \cdot 36}{2^2} \\ & \approx 11.116332802\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 471744 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $43$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.6.0.6 | $6$ |
| $3$ | 3B | 9.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 21672 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 43 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 21637 & 36 \\ 21636 & 37 \end{array}\right),\left(\begin{array}{rr} 16255 & 1548 \\ 14878 & 15481 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 18143 & 0 \\ 0 & 21671 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21415 & 1548 \\ 6794 & 11095 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 3240 & 6139 \end{array}\right),\left(\begin{array}{rr} 6580 & 6579 \\ 18705 & 7354 \end{array}\right),\left(\begin{array}{rr} 17029 & 1548 \\ 18232 & 15653 \end{array}\right)$.
The torsion field $K:=\Q(E[21672])$ is a degree-$46506533584896$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/21672\Z)$.
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 | $4$ | \( 1849 = 43^{2} \) |
| $3$ | good | $2$ | \( 12943 = 7 \cdot 43^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 3698 = 2 \cdot 43^{2} \) |
| $43$ | additive | $926$ | \( 14 = 2 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 25886a
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a5, its twist by $-43$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{129}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-86 +86 \sqrt{-7}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{129})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.6.3757412010681.1 | \(\Z/18\Z\) | not in database |
| $6$ | 6.0.139163407803.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.702635044765696.9 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.537954956148736.48 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.680849241375744.23 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 |
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 | ss | nonsplit | ss | ord | ord | ord | ss | ord | ord | ord | ord | add | ord |
| $\lambda$-invariant(s) | 2 | 1 | 1,3 | 1 | 1,3 | 1 | 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 | - | 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.