Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-148x+644\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-148xz^2+644z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-12015x+505494\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(20, 78\right) \)
Integral points
\( \left(7, 0\right) \), \((8,\pm 6)\), \((20,\pm 78)\)
Invariants
Conductor: | \( 156 \) | = | $2^{2} \cdot 3 \cdot 13$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $1168128 $ | = | $2^{8} \cdot 3^{3} \cdot 13^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1409938000}{4563} \) | = | $2^{4} \cdot 3^{-3} \cdot 5^{3} \cdot 13^{-2} \cdot 89^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.030152328504919680961576991061\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.43194579186837719198324442324\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9334220298106218\dots$ | |||
Szpiro ratio: | $5.269851056151717\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $2.7522991872794050113370038521\dots$ | comment: Real Period
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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Tamagawa product: | $ 18 $ = $ 3\cdot3\cdot2 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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Torsion order: | $6$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 1.3761495936397025056685019261 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 1.376149594 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.752299 \cdot 1.000000 \cdot 18}{6^2} \approx 1.376149594$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 24 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
$3$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$13$ | $2$ | $I_{2}$ | 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 | 2.3.0.1 |
$3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 156 = 2^{2} \cdot 3 \cdot 13 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 66 & 25 \\ 65 & 14 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 106 & 147 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 145 & 12 \\ 144 & 13 \end{array}\right),\left(\begin{array}{rr} 145 & 12 \\ 90 & 73 \end{array}\right),\left(\begin{array}{rr} 110 & 11 \\ 93 & 136 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[156])$ is a degree-$1257984$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/156\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 156.b
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.2.12.1-2028.1-a6 |
$4$ | 4.4.8112.1 | \(\Z/12\Z\) | Not in database |
$6$ | 6.0.12338352.2 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$8$ | 8.0.32296402944.14 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$8$ | 8.8.9475854336.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$9$ | 9.3.26228424500928.2 | \(\Z/18\Z\) | Not in database |
$12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.6.76079582407162943727891464060928.2 | \(\Z/2\Z \oplus \Z/18\Z\) | 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.
Iwasawa invariants
$p$ | 2 | 3 | 13 |
---|---|---|---|
Reduction type | add | split | 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.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.