Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+42268x+391839\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+42268xz^2+391839z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+676293x+25754006\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(47, 1551)$ | $0.50644262870348820531108033620$ | $\infty$ |
| $(-37/4, 33/8)$ | $0$ | $2$ |
Integral points
\( \left(47, 1551\right) \), \( \left(47, -1599\right) \), \( \left(147, 3051\right) \), \( \left(147, -3199\right) \), \( \left(263, 5313\right) \), \( \left(263, -5577\right) \), \( \left(747, 20801\right) \), \( \left(747, -21549\right) \)
Invariants
| Conductor: | $N$ | = | \( 6930 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-4903042134375000$ | = | $-1 \cdot 2^{3} \cdot 3^{7} \cdot 5^{8} \cdot 7^{2} \cdot 11^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{11456208593737991}{6725709375000} \) | = | $2^{-3} \cdot 3^{-1} \cdot 5^{-8} \cdot 7^{-2} \cdot 11^{-4} \cdot 225431^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7001120770507326932308582367$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1508059327166778475332356182$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0150912911963035$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.926603147137274$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.50644262870348820531108033620$ |
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| Real period: | $\Omega$ | ≈ | $0.26243998356700128923567036839$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 3\cdot2\cdot2^{3}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.3797181674194742419833679308 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.379718167 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.262440 \cdot 0.506443 \cdot 192}{2^2} \\ & \approx 6.379718167\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49152 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $5$ | $8$ | $I_{8}$ | split multiplicative | -1 | 1 | 8 | 8 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 |
|---|---|---|
| $2$ | 2B | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 172 & 263 \\ 65 & 258 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 232 & 107 \\ 169 & 198 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 235 & 232 \\ 122 & 237 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 258 & 259 \end{array}\right),\left(\begin{array}{rr} 257 & 8 \\ 256 & 9 \end{array}\right),\left(\begin{array}{rr} 145 & 8 \\ 52 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$20275200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 385 = 5 \cdot 7 \cdot 11 \) |
| $5$ | split multiplicative | $6$ | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
| $11$ | nonsplit 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 and 4.
Its isogeny class 6930bi
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2310c4, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.2709504.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.29365647704064.50 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.4974113193984.48 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.104966417977344.87 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.3892034846266875.1 | \(\Z/6\Z\) | 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 |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | add | split | split | nonsplit | ord | ord | ss | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | - | 2 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1,3 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | - | 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.