Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
|
\(y^2+xy=x^3+x^2-85x-473\)
|
(homogenize, simplify) |
|
\(y^2z+xyz=x^3+x^2z-85xz^2-473z^3\)
|
(dehomogenize, simplify) |
|
\(y^2=x^3-110835x-20409138\)
|
(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{43}{4}, -\frac{43}{8}\right) \)
Integral points
None
Invariants
|
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
|
|||
| Conductor: | \( 114 \) | = | $2 \cdot 3 \cdot 19$ |
|
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
|
|||
| Discriminant: | $-42633378 $ | = | $-1 \cdot 2 \cdot 3^{10} \cdot 19^{2} $ |
|
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
|
|||
| j-invariant: | \( -\frac{69173457625}{42633378} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-10} \cdot 5^{3} \cdot 19^{-2} \cdot 821^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.16588295511894989169631062847\dots$ | ||
| Stable Faltings height: | $0.16588295511894989169631062847\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
| |||
| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
| |||
| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
| |||
| Real period: | $0.76357205651559192806870953186\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
| |||
| Tamagawa product: | $ 4 $ = $ 1\cdot2\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
| |||
| Torsion order: | $2$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
| |||
| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
| |||
| Special value: | $ L(E,1) $ ≈ $ 0.76357205651559192806870953186 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
|
sage: E.modular_degree()
magma: ModularDegree(E);
|
|||
| Modular degree: | 40 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $2$ | $I_{10}$ | Non-split multiplicative | 1 | 1 | 10 | 10 |
| $19$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
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 | 8.6.0.5 |
The image of the adelic Galois representation has level $456$, index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 305 & 4 \\ 154 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 4 \\ 194 & 9 \end{array}\right),\left(\begin{array}{rr} 172 & 289 \\ 57 & 400 \end{array}\right),\left(\begin{array}{rr} 453 & 4 \\ 452 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 227 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 19 |
|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split |
| $\lambda$-invariant(s) | 0 | 0 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 114.a
consists of 2 curves linked by isogenies of
degree 2.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.8.1-6498.5-a3 |
| $4$ | 4.2.103968.2 | \(\Z/4\Z\) | Not in database |
| $8$ | 8.0.13627293696.9 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.0.691798081536.14 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | 8.2.369375586992.1 | \(\Z/6\Z\) | Not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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.