Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2+4161x+3178\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z+4161xz^2+3178z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+5392224x+212992848\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1974364/210681, 19982765431/96702579)$ | $12.129074238432003713635130373$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 118579 \) | = | $19 \cdot 79^{2}$ |
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| Discriminant: | $\Delta$ | = | $-4618661654899$ | = | $-1 \cdot 19 \cdot 79^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{32768}{19} \) | = | $2^{15} \cdot 19^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1195507945567188592113224174$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0651731316767918878751503533$ |
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| $abc$ quality: | $Q$ | ≈ | $1.3175706029138485$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.133856541234594$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $12.129074238432003713635130373$ |
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| Real period: | $\Omega$ | ≈ | $0.46433417159249136813799229502$ |
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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)$ | ≈ | $11.263887277372305148031089008 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 11.263887277 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.464334 \cdot 12.129074 \cdot 2}{1^2} \\ & \approx 11.263887277\end{aligned}$$
Modular invariants
Modular form 118579.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 164736 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $79$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 81054 = 2 \cdot 3^{3} \cdot 19 \cdot 79 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 31 & 36 \\ 75232 & 74293 \end{array}\right),\left(\begin{array}{rr} 2051 & 0 \\ 0 & 81053 \end{array}\right),\left(\begin{array}{rr} 47164 & 9243 \\ 59171 & 80818 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 70943 & 50955 \\ 35155 & 53800 \end{array}\right),\left(\begin{array}{rr} 81001 & 54 \\ 81000 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right)$.
The torsion field $K:=\Q(E[81054])$ is a degree-$6902277479884800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/81054\Z)$.
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 |
|---|---|---|---|
| $19$ | split multiplicative | $20$ | \( 6241 = 79^{2} \) |
| $79$ | additive | $3122$ | \( 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 118579c
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 19a3, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-79}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1734840059013.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.64253335519.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.2847793264.2 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.23094282812104183689.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.7720500854944980479727858859538343469535232.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.392242079710663033060400287534336405504.1 | \(\Z/18\Z\) | 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 | ss | ord | ord | ord | ord | ord | ord | split | ss | ord | ord | ord | ord | ord | ord | add |
| $\lambda$-invariant(s) | 8,11 | 3 | 1 | 1 | 1 | 1 | 1 | 2 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.