Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-361371x+83463910\)
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(homogenize, simplify) |
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\(y^2z=x^3-361371xz^2+83463910z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-361371x+83463910\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(398, 1638\right) \) | $1.4222147340654639884442290942$ | $\infty$ |
| \( \left(335, 0\right) \) | $0$ | $2$ |
| \( \left(359, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([398:1638:1]\) | $1.4222147340654639884442290942$ | $\infty$ |
| \([335:0:1]\) | $0$ | $2$ |
| \([359:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(398, 1638\right) \) | $1.4222147340654639884442290942$ | $\infty$ |
| \( \left(335, 0\right) \) | $0$ | $2$ |
| \( \left(359, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-694, 0\right) \), \((-57,\pm 10192)\), \( \left(335, 0\right) \), \( \left(359, 0\right) \), \((398,\pm 1638)\), \((1007,\pm 27216)\)
\([-694:0:1]\), \([-57:\pm 10192:1]\), \([335:0:1]\), \([359:0:1]\), \([398:\pm 1638:1]\), \([1007:\pm 27216:1]\)
\( \left(-694, 0\right) \), \((-57,\pm 10192)\), \( \left(335, 0\right) \), \( \left(359, 0\right) \), \((398,\pm 1638)\), \((1007,\pm 27216)\)
Invariants
| Conductor: | $N$ | = | \( 6552 \) | = | $2^{3} \cdot 3^{2} \cdot 7 \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $10820067198280704$ | = | $2^{10} \cdot 3^{12} \cdot 7^{6} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{6991270724335972}{14494474449} \) | = | $2^{2} \cdot 3^{-6} \cdot 7^{-6} \cdot 13^{-2} \cdot 163^{3} \cdot 739^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9624243136602431171933904251$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.83549551885956718031474103876$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9873046363895048$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.6906331082204185$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4222147340654639884442290942$ |
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| Real period: | $\Omega$ | ≈ | $0.40555948737812297788438085653$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.4607560709352184200377491227 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.460756071 \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 0.405559 \cdot 1.422215 \cdot 96}{4^2} \\ & \approx 3.460756071\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64512 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $7$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1683 & 2 \\ 334 & 2183 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 549 & 4 \\ 1094 & 3 \end{array}\right),\left(\begin{array}{rr} 2181 & 2182 \\ 730 & 1 \end{array}\right),\left(\begin{array}{rr} 2181 & 4 \\ 2180 & 5 \end{array}\right),\left(\begin{array}{rr} 1093 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 1249 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2184])$ is a degree-$81155063808$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2184\Z)$.
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 | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $6$ | \( 104 = 2^{3} \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 6552j
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2184l2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{6}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-13}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.63962016768.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | 16.0.132513778481397717569346994176.1 | \(\Z/4\Z \oplus \Z/4\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 3 | 2 | 1 | 5 | 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 |
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.