Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-2994759x+1995489513\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-2994759xz^2+1995489513z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-47916147x+127663412686\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1044, 1935)$ | $2.5503935441240096438128675703$ | $\infty$ |
| $(1002, -501)$ | $0$ | $2$ |
| $(972, 999)$ | $0$ | $6$ |
Integral points
\( \left(-1998, 999\right) \), \( \left(-1833, 37299\right) \), \( \left(-1833, -35466\right) \), \( \left(202, 37299\right) \), \( \left(202, -37501\right) \), \( \left(477, 25749\right) \), \( \left(477, -26226\right) \), \( \left(664, 16971\right) \), \( \left(664, -17635\right) \), \( \left(972, 999\right) \), \( \left(972, -1971\right) \), \( \left(1002, -501\right) \), \( \left(1027, 999\right) \), \( \left(1027, -2026\right) \), \( \left(1044, 1935\right) \), \( \left(1044, -2979\right) \), \( \left(1377, 21249\right) \), \( \left(1377, -22626\right) \), \( \left(1632, 37299\right) \), \( \left(1632, -38931\right) \), \( \left(52452, 11979999\right) \), \( \left(52452, -12032451\right) \)
Invariants
| Conductor: | $N$ | = | \( 6930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $35596085895562500$ | = | $2^{2} \cdot 3^{8} \cdot 5^{6} \cdot 7^{2} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4074571110566294433649}{48828650062500} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-2} \cdot 11^{-6} \cdot 37^{3} \cdot 43^{3} \cdot 10039^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3251541852555119559139602147$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7758480409214571102163375962$ |
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| $abc$ quality: | $Q$ | ≈ | $1.01113946906003$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.371910761573034$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5503935441240096438128675703$ |
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| Real period: | $\Omega$ | ≈ | $0.33315576776726077978845092890$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 576 $ = $ 2\cdot2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $12$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.3987132772051988617275654828 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.398713277 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.333156 \cdot 2.550394 \cdot 576}{12^2} \\ & \approx 3.398713277\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 165888 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 2521 & 12 \\ 5886 & 73 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 9224 & 9233 \end{array}\right),\left(\begin{array}{rr} 6937 & 6 \\ 9192 & 9199 \end{array}\right),\left(\begin{array}{rr} 4627 & 6 \\ 9234 & 9235 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 9229 & 12 \\ 9228 & 13 \end{array}\right),\left(\begin{array}{rr} 7399 & 6 \\ 5538 & 9235 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5281 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3079 & 9228 \\ 7694 & 9167 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 14 = 2 \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 6930o
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310t6, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-11}, \sqrt{-30})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{11}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{21}, \sqrt{30})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.6805279152.4 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $9$ | 9.3.15258333411304656750000.2 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | 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 | nonsplit | add | split | split | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | - | 2 | 2 | 2 | 1 | 1 | 3 | 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 |
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.