Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-6653x-36089161\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-6653xz^2-36089161z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8622315x-1683646552602\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(8143/9, 701914/27)$ | $6.4808576578192466252533348265$ | $\infty$ |
| $(1347/4, -1351/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 19074 \) | = | $2 \cdot 3 \cdot 11 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-562543412332794948$ | = | $-1 \cdot 2^{2} \cdot 3^{4} \cdot 11^{4} \cdot 17^{9} $ |
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| j-invariant: | $j$ | = | \( -\frac{274625}{4743684} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-4} \cdot 5^{3} \cdot 11^{-4} \cdot 13^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0848394794368273771806352883$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.040070528605334683006515675106$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0770442522930508$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.9031654717541535$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.4808576578192466252533348265$ |
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| Real period: | $\Omega$ | ≈ | $0.13276640864900433332687254190$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.8835215695524735304027532945 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.883521570 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.132766 \cdot 6.480858 \cdot 32}{2^2} \\ & \approx 6.883521570\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 208896 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $17$ | $2$ | $III^{*}$ | additive | 1 | 2 | 9 | 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 | 4.12.0.11 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4488 = 2^{3} \cdot 3 \cdot 11 \cdot 17 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 8 \\ 4480 & 4467 \end{array}\right),\left(\begin{array}{rr} 4481 & 8 \\ 4480 & 9 \end{array}\right),\left(\begin{array}{rr} 2249 & 1124 \\ 1144 & 2253 \end{array}\right),\left(\begin{array}{rr} 3370 & 3369 \\ 21 & 3382 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 8 \\ 48 & 77 \end{array}\right),\left(\begin{array}{rr} 1854 & 19 \\ 1577 & 4466 \end{array}\right),\left(\begin{array}{rr} 409 & 8 \\ 1636 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2993 & 8 \\ 2996 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[4488])$ is a degree-$1588278067200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4488\Z)$.
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 | $4$ | \( 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 6358 = 2 \cdot 11 \cdot 17^{2} \) |
| $11$ | split multiplicative | $12$ | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| $17$ | additive | $98$ | \( 66 = 2 \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 19074s
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 19074z2, its twist by $17$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.2.19652.1 | \(\Z/4\Z\) | not in database |
| $4$ | 4.0.78608.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6179217664.3 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 |
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 | nonsplit | ss | ss | split | ord | add | ss | ord | ord | ss | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 6 | 1 | 1,3 | 1,1 | 2 | 1 | - | 1,1 | 1 | 1 | 1,1 | 3 | 1,1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 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.