Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+146x+1749\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+146xz^2+1749z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+189189x+81033750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(4, 47)$ | $1.0793128183525912286713702981$ | $\infty$ |
| $(-33/4, 33/8)$ | $0$ | $2$ |
Integral points
\( \left(4, 47\right) \), \( \left(4, -51\right) \), \( \left(20, 103\right) \), \( \left(20, -123\right) \)
Invariants
| Conductor: | $N$ | = | \( 5537 \) | = | $7^{2} \cdot 113$ |
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| Discriminant: | $\Delta$ | = | $-1502260081$ | = | $-1 \cdot 7^{6} \cdot 113^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2924207}{12769} \) | = | $11^{3} \cdot 13^{3} \cdot 113^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.44335540841060696924208464911$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.52959966611704968331059172261$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1245004508528507$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.299930121933208$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0793128183525912286713702981$ |
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| Real period: | $\Omega$ | ≈ | $1.0801514212953626421140396916$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.1658212747658549803776885033 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.165821275 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.080151 \cdot 1.079313 \cdot 4}{2^2} \\ & \approx 1.165821275\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2304 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $113$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6328 = 2^{3} \cdot 7 \cdot 113 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 8 \\ 10 & 27 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6105 & 910 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 3165 & 5432 \\ 4522 & 4537 \end{array}\right),\left(\begin{array}{rr} 6321 & 8 \\ 6320 & 9 \end{array}\right),\left(\begin{array}{rr} 4519 & 0 \\ 0 & 6327 \end{array}\right),\left(\begin{array}{rr} 1583 & 5432 \\ 567 & 4537 \end{array}\right)$.
The torsion field $K:=\Q(E[6328])$ is a degree-$10424610717696$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6328\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 49 = 7^{2} \) |
| $7$ | additive | $26$ | \( 113 \) |
| $113$ | split multiplicative | $114$ | \( 49 = 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 5537.a
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 113.a2, its twist by $-7$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.2502724.1 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.2009226870784.23 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.100218038722816.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 113 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | ord | ord | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | 8 | 1 | 1 | - | 3,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-invariant(s) | 1 | 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.