Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-1062x+13590\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-1062xz^2+13590z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-16995x+852766\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(19, -9\right) \)
Integral points
\( \left(19, -9\right) \), \( \left(19, -10\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 378 \) | = | $2 \cdot 3^{3} \cdot 7$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-3402 $ | = | $-1 \cdot 2 \cdot 3^{5} \cdot 7 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{545407363875}{14} \) | = | $-1 \cdot 2^{-1} \cdot 3 \cdot 5^{3} \cdot 7^{-1} \cdot 11^{3} \cdot 103^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.19388385769632813006906179683\dots$ | ||
| Stable Faltings height: | $-0.26387126258205090801229038522\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $3.2456774249956641238240628560\dots$ | ||
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: | $ 3 $ = $ 1\cdot3\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $3$ | ||
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) $ ≈ $ 1.0818924749985547079413542853 $ | ||
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: | 108 | ||
| $ \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)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $3$ | $IV$ | Additive | 1 | 3 | 5 | 0 |
| $7$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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.1.1 | 9.24.0.1 |
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 7 |
|---|---|---|---|
| Reduction type | nonsplit | add | split |
| $\lambda$-invariant(s) | 1 | - | 1 |
| $\mu$-invariant(s) | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
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 and 9.
Its isogeny class 378b
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.1512.1 | \(\Z/6\Z\) | Not in database |
| $3$ | \(\Q(\zeta_{7})^+\) | \(\Z/9\Z\) | Not in database |
| $6$ | 6.0.384072192.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.6805279152.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $6$ | 6.0.138883248.16 | \(\Z/9\Z\) | Not in database |
| $9$ | 9.3.8299415996928.1 | \(\Z/18\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/12\Z\) | Not in database |
| $18$ | 18.0.315164892649262158461997559808.4 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
| $18$ | 18.0.20494572895025913969633989670960365568.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.0.8535848769273600153950016522682368.1 | \(\Z/18\Z\) | Not in database |
| $18$ | 18.0.6665409440369750708186945421312.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.