Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2+423612946x+8846527401901\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z+423612946xz^2+8846527401901z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+6777807133x+566184531528798\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 118579 \) | = | $19 \cdot 79^{2}$ |
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| Discriminant: | $\Delta$ | = | $-38674635582769950658581013501$ | = | $-1 \cdot 19^{9} \cdot 79^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{70143520960521}{322687697779} \) | = | $3^{3} \cdot 19^{-9} \cdot 59^{3} \cdot 233^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1678118583838646520366019997$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.89072596903359853140689284359$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0584726267565254$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.260649351019175$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.026107870576021607720245430268$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.46994167036838893896441774482 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 0.469941670 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.026108 \cdot 1.000000 \cdot 2}{1^2} \\ & \approx 0.469941670\end{aligned}$$
Modular invariants
Modular form 118579.2.a.b
For more coefficients, see the Downloads section to the right.
| Modular degree: | 57335040 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $19$ | $1$ | $I_{9}$ | nonsplit multiplicative | 1 | 1 | 9 | 9 |
| $79$ | $2$ | $III^{*}$ | 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 |
|---|---|---|
| $3$ | 3Nn | 9.27.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 54036 = 2^{2} \cdot 3^{2} \cdot 19 \cdot 79 \), index $108$, genus $7$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 3 & 16 \\ 53852 & 53055 \end{array}\right),\left(\begin{array}{rr} 27019 & 18 \\ 27027 & 163 \end{array}\right),\left(\begin{array}{rr} 43781 & 46 \\ 15067 & 175 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 25522 & 37571 \\ 13123 & 5455 \end{array}\right),\left(\begin{array}{rr} 8533 & 18 \\ 22761 & 163 \end{array}\right),\left(\begin{array}{rr} 54019 & 18 \\ 54018 & 19 \end{array}\right)$.
The torsion field $K:=\Q(E[54036])$ is a degree-$16360954026393600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/54036\Z)$.
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 |
|---|---|---|---|
| $3$ | good | $2$ | \( 6241 = 79^{2} \) |
| $19$ | nonsplit multiplicative | $20$ | \( 6241 = 79^{2} \) |
| $79$ | additive | $1640$ | \( 19 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 118579b consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 118579a1, its twist by $-79$.
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 |
|---|---|---|---|
| $3$ | 3.1.6004.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.216432288064.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.531632265224427.1 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $16$ | 16.0.282632865427655493724677478329.1 | \(\Z/3\Z \oplus \Z/3\Z\) | 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 | 79 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | ss | ord | ss | ord | ord | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord | add |
| $\lambda$-invariant(s) | 5 | 2,2 | 6 | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - |
| $\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.