Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-1546933295x-23418435858105\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-1546933295xz^2-23418435858105z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2004825550347x-1092604528919095866\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{17802987}{784}, \frac{249244689}{21952}\right) \) | $9.6312314638802287071300423167$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-498483636:249244689:21952]\) | $9.6312314638802287071300423167$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-\frac{160226295}{196}, \frac{77517}{2744}\right) \) | $9.6312314638802287071300423167$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 32370 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 83$ |
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| Minimal Discriminant: | $\Delta$ | = | $10583059885954530$ | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 83^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{409391171678522880998024667047281}{10583059885954530} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-1} \cdot 13^{-1} \cdot 83^{-7} \cdot 199^{3} \cdot 8693^{3} \cdot 42923^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5174050487602946826263405144$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.5174050487602946826263405144$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0291985959437848$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.230844043448585$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $9.6312314638802287071300423167$ |
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| Real period: | $\Omega$ | ≈ | $0.024073079935701455525477305243$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $11.360816840552352881555330864 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $49$ = $7^2$ (rounded) |
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BSD formula
$$\begin{aligned} 11.360816841 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{49 \cdot 0.024073 \cdot 9.631231 \cdot 1}{1^2} \\ & \approx 11.360816841\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6991712 |
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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 semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $83$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
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 |
|---|---|---|---|
| $7$ | 7B.1.3 | 7.48.0.5 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 906360 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 83 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 226591 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 453181 & 14 \\ 453187 & 99 \end{array}\right),\left(\begin{array}{rr} 578761 & 14 \\ 425887 & 99 \end{array}\right),\left(\begin{array}{rr} 543817 & 14 \\ 181279 & 99 \end{array}\right),\left(\begin{array}{rr} 906347 & 14 \\ 906346 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 69721 & 14 \\ 488047 & 99 \end{array}\right),\left(\begin{array}{rr} 226593 & 258968 \\ 453166 & 614993 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 604241 & 14 \\ 604247 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[906360])$ is a degree-$913086556114004213760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/906360\Z)$.
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 | $4$ | \( 16185 = 3 \cdot 5 \cdot 13 \cdot 83 \) |
| $3$ | split multiplicative | $4$ | \( 10790 = 2 \cdot 5 \cdot 13 \cdot 83 \) |
| $5$ | split multiplicative | $6$ | \( 6474 = 2 \cdot 3 \cdot 13 \cdot 83 \) |
| $7$ | good | $2$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 2490 = 2 \cdot 3 \cdot 5 \cdot 83 \) |
| $83$ | nonsplit multiplicative | $84$ | \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 32370.bk
consists of 2 curves linked by isogenies of
degree 7.
Twists
This elliptic curve is its own minimal quadratic twist.
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 |
|---|---|---|---|
| $3$ | 3.3.129480.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.2170741315392000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | not in database |
| $7$ | 7.1.2897836793165223000000.1 | \(\Z/7\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.22371069374616506378182176937754161152000000.1 | \(\Z/14\Z\) | not in database |
| $21$ | 21.3.4219421968712344997259665288738671312971805871603752938250870947840000000000000000000.1 | \(\Z/14\Z\) | 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 | 83 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | split | split | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 2 | 8 | 2 | 5 | 1 | 4 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.