Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-219x+4106\)
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(homogenize, simplify) |
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\(y^2z=x^3-219xz^2+4106z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-219x+4106\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 216\right) \) | $0.34925917399531236341535155670$ | $\infty$ |
| \( \left(-11, 72\right) \) | $0.70106575052915844523215600272$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([37:216:1]\) | $0.34925917399531236341535155670$ | $\infty$ |
| \([-11:72:1]\) | $0.70106575052915844523215600272$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37, 216\right) \) | $0.34925917399531236341535155670$ | $\infty$ |
| \( \left(-11, 72\right) \) | $0.70106575052915844523215600272$ | $\infty$ |
Integral points
\((-17,\pm 54)\), \((-11,\pm 72)\), \((5,\pm 56)\), \((7,\pm 54)\), \((10,\pm 54)\), \((23,\pm 106)\), \((37,\pm 216)\), \((53,\pm 376)\), \((70,\pm 576)\), \((469,\pm 10152)\), \((1279,\pm 45738)\), \((3013,\pm 165384)\)
\([-17:\pm 54:1]\), \([-11:\pm 72:1]\), \([5:\pm 56:1]\), \([7:\pm 54:1]\), \([10:\pm 54:1]\), \([23:\pm 106:1]\), \([37:\pm 216:1]\), \([53:\pm 376:1]\), \([70:\pm 576:1]\), \([469:\pm 10152:1]\), \([1279:\pm 45738:1]\), \([3013:\pm 165384:1]\)
\((-17,\pm 54)\), \((-11,\pm 72)\), \((5,\pm 56)\), \((7,\pm 54)\), \((10,\pm 54)\), \((23,\pm 106)\), \((37,\pm 216)\), \((53,\pm 376)\), \((70,\pm 576)\), \((469,\pm 10152)\), \((1279,\pm 45738)\), \((3013,\pm 165384)\)
Invariants
| Conductor: | $N$ | = | \( 5904 \) | = | $2^{4} \cdot 3^{2} \cdot 41$ |
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| Minimal Discriminant: | $\Delta$ | = | $-6610968576$ | = | $-1 \cdot 2^{13} \cdot 3^{9} \cdot 41 $ |
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| j-invariant: | $j$ | = | \( -\frac{389017}{2214} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-3} \cdot 41^{-1} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.56824812348767583439687934507$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.67420520140632432071797539485$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8755235934443263$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4737090962124366$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.23122685214454472784689594595$ |
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| Real period: | $\Omega$ | ≈ | $1.1532445301726631528058264496$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.2665776394358293445506430843 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.266577639 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.153245 \cdot 0.231227 \cdot 16}{1^2} \\ & \approx 4.266577639\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3840 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $41$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 984 = 2^{3} \cdot 3 \cdot 41 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 493 & 6 \\ 495 & 19 \end{array}\right),\left(\begin{array}{rr} 737 & 978 \\ 0 & 983 \end{array}\right),\left(\begin{array}{rr} 457 & 6 \\ 387 & 19 \end{array}\right),\left(\begin{array}{rr} 450 & 527 \\ 653 & 186 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 979 & 6 \\ 978 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[984])$ is a degree-$12695961600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/984\Z)$.
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 | $4$ | \( 369 = 3^{2} \cdot 41 \) |
| $3$ | additive | $2$ | \( 656 = 2^{4} \cdot 41 \) |
| $41$ | split multiplicative | $42$ | \( 144 = 2^{4} \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 5904v
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 246f1, its twist by $12$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.984.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.952763904.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.6510553344.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.2.46476288.1 | \(\Z/6\Z\) | 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 |
| $18$ | 18.6.186179630527837374767087685344428032.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.4275273318331610444736997685999960064.1 | \(\Z/6\Z\) | 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 | add | add | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord | split | ord | ord |
| $\lambda$-invariant(s) | - | - | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 3 | 2 | 2 |
| $\mu$-invariant(s) | - | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.