Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-66810x+6640963\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-66810xz^2+6640963z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-86585139x+310100536782\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(23, 2250)$ | $4.3714751915716888414426280188$ | $\infty$ |
| $(599/4, -603/8)$ | $0$ | $2$ |
Integral points
\( \left(23, 2250\right) \), \( \left(23, -2274\right) \)
Invariants
| Conductor: | $N$ | = | \( 37479 \) | = | $3 \cdot 13 \cdot 31^{2}$ |
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| Discriminant: | $\Delta$ | = | $934541376093$ | = | $3^{4} \cdot 13 \cdot 31^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{37159393753}{1053} \) | = | $3^{-4} \cdot 13^{-1} \cdot 47^{3} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3986271401141856700158051451$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.31836646212838745294877701717$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1161605159157866$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.267412220594493$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3714751915716888414426280188$ |
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| Real period: | $\Omega$ | ≈ | $0.82136688431563754157462877857$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.1811699159286856176709373632 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.181169916 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.821367 \cdot 4.371475 \cdot 8}{2^2} \\ & \approx 7.181169916\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 120960 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $13$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $31$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9672 = 2^{3} \cdot 3 \cdot 13 \cdot 31 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3163 & 3162 \\ 8866 & 4651 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9666 & 9667 \end{array}\right),\left(\begin{array}{rr} 9359 & 0 \\ 0 & 9671 \end{array}\right),\left(\begin{array}{rr} 6449 & 8432 \\ 5828 & 4713 \end{array}\right),\left(\begin{array}{rr} 2140 & 4681 \\ 527 & 4062 \end{array}\right),\left(\begin{array}{rr} 3163 & 3472 \\ 4898 & 2853 \end{array}\right),\left(\begin{array}{rr} 9665 & 8 \\ 9664 & 9 \end{array}\right)$.
The torsion field $K:=\Q(E[9672])$ is a degree-$35940099686400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9672\Z)$.
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 | $2$ | \( 12493 = 13 \cdot 31^{2} \) |
| $3$ | split multiplicative | $4$ | \( 12493 = 13 \cdot 31^{2} \) |
| $13$ | nonsplit multiplicative | $14$ | \( 2883 = 3 \cdot 31^{2} \) |
| $31$ | additive | $482$ | \( 39 = 3 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 37479.g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 39.a1, its twist by $-31$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{31}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{403}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{13}, \sqrt{31})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1141160825469184.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | ord | split | ord | ord | ord | nonsplit | ord | ss | ss | ord | add | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 16 | 2 | 1 | 1 | 1 | 1 | 1 | 1,1 | 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 | 0,0 |
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.