Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-122x-4948\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-122xz^2-4948z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-157491x-230369778\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Torsion generators
\( \left(30, 121\right) \)
Integral points
\( \left(30, 121\right) \), \( \left(30, -152\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 546 \) | = | $2 \cdot 3 \cdot 7 \cdot 13$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-10417365504 $ | = | $-1 \cdot 2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 13^{3} $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{198461344537}{10417365504} \) | = | $-1 \cdot 2^{-9} \cdot 3^{-3} \cdot 7^{-3} \cdot 13^{-3} \cdot 19^{3} \cdot 307^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $0.60215319124588882820324886666\dots$ | ||
| Stable Faltings height: | $0.60215319124588882820324886666\dots$ | ||
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: 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: | $0.56365221639777019227436627275\dots$ | ||
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: | $ 27 $ = $ 1\cdot3\cdot3\cdot3 $ | ||
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) $ ≈ $ 1.6909566491933105768230988183 $ | ||
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: | 648 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{9}$ | Non-split multiplicative | 1 | 1 | 9 | 9 |
| $3$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $13$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|
| $3$ | 3Cs.1.1 | 3.24.0.1 |
The image of the adelic Galois representation has level $6552$, index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 10 & 9 \\ 3267 & 6544 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3278 & 4923 \\ 1593 & 5986 \end{array}\right),\left(\begin{array}{rr} 2812 & 9 \\ 1863 & 6532 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 6064 & 9 \\ 3663 & 76 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3286 & 9 \\ 4905 & 6544 \end{array}\right),\left(\begin{array}{rr} 6535 & 18 \\ 6534 & 19 \end{array}\right)$
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 7 | 13 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | split | split |
| $\lambda$-invariant(s) | 2 | 9 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 546.d
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z \oplus \Z/3\Z\) | 2.0.3.1-99372.5-k3 |
| $3$ | 3.1.2184.1 | \(\Z/6\Z\) | Not in database |
| $6$ | 6.0.10417365504.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.14309568.3 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $9$ | 9.3.8148303085695869187.16 | \(\Z/9\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.0.3156819565980502545707572443.1 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
| $18$ | 18.0.2459068265791141604143011767237631147.18 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
| $18$ | 18.0.1792660765761742229420255578316233106163.1 | \(\Z/3\Z \oplus \Z/9\Z\) | 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.