Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2+3167x+11119\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+3167xz^2+11119z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+4104000x+469530000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(\frac{181}{4}, \frac{3989}{8}\right)\)
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| $\hat{h}(P)$ | ≈ | $2.2524072734962648026370090497$ |
Integral points
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 825 \) | = | $3 \cdot 5^{2} \cdot 11$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-2076048046875 $ | = | $-1 \cdot 3 \cdot 5^{8} \cdot 11^{6} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( \frac{8990228480}{5314683} \) | = | $2^{18} \cdot 3^{-1} \cdot 5 \cdot 11^{-6} \cdot 19^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.0528142912586010139522802288\dots$ | ||
| Stable Faltings height: | $-0.020144317030799235781559326684\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $1$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $2.2524072734962648026370090497\dots$ | ||
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);
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| Real period: | $0.50308198258933344563298817670\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 2 $ = $ 1\cdot1\cdot2 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L'(E,1) $ ≈ $ 2.2662910334982718119316468866 $ | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 1080 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $1$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
| $11$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
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.1.2 | 3.8.0.2 |
The image of the adelic Galois representation has level $6$, index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 2 & 3 \\ 1 & 4 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 3 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 2 & 5 \end{array}\right)$
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | split | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 1,2 | 2 | - | 1 | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 0,0 | 1 | - | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 825.c
consists of 2 curves linked by isogenies of
degree 3.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | Not in database |
| $3$ | 3.1.6075.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.110716875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $9$ | 9.1.43046721000000.3 | \(\Z/6\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | Not in database |
| $18$ | 18.0.473272391221478207226757272216796875.2 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.0.5559060566555523000000000000.3 | \(\Z/6\Z \oplus \Z/6\Z\) | 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.