Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-764x-16264\)
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(homogenize, simplify) |
\(y^2z+xyz=x^3-764xz^2-16264z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-990171x-755842698\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 338 \) | = | $2 \cdot 13^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-84835994984 $ | = | $-1 \cdot 2^{3} \cdot 13^{9} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{10218313}{17576} \) | = | $-1 \cdot 2^{-3} \cdot 7^{3} \cdot 13^{-3} \cdot 31^{3}$ | 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: | $0.78755496053009017810233161301\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.49491971820067818992441210777\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.947172997962475\dots$ | |||
Szpiro ratio: | $5.651461482474274\dots$ |
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: | $0.42898570444899263812709637839\dots$ | 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: | $ 6 $ = $ 3\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: | $1$ | 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) $ ≈ $ 2.5739142266939558287625782703 $ | 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
$\displaystyle 2.573914227 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.428986 \cdot 1.000000 \cdot 6}{1^2} \approx 2.573914227$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 336 | 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 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
$13$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 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 | 3.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 923 & 918 \\ 783 & 425 \end{array}\right),\left(\begin{array}{rr} 469 & 18 \\ 477 & 163 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 713 & 486 \\ 306 & 541 \end{array}\right),\left(\begin{array}{rr} 919 & 18 \\ 918 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 703 & 18 \\ 711 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[936])$ is a degree-$1086898176$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/936\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | split multiplicative | $4$ | \( 169 = 13^{2} \) |
$3$ | good | $2$ | \( 169 = 13^{2} \) |
$13$ | additive | $98$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 338.f
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 26.a2, its twist by $13$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{13}) \) | \(\Z/3\Z\) | 2.2.13.1-52.1-a2 |
$2$ | \(\Q(\sqrt{-39}) \) | \(\Z/3\Z\) | not in database |
$3$ | 3.1.104.1 | \(\Z/2\Z\) | not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{13})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
$6$ | 6.0.1124864.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$6$ | 6.2.140608.1 | \(\Z/6\Z\) | not in database |
$6$ | 6.0.3796416.1 | \(\Z/6\Z\) | not in database |
$12$ | 12.2.8421963387109376.7 | \(\Z/4\Z\) | not in database |
$12$ | 12.0.14412774445056.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$12$ | 12.0.1265319018496.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$12$ | 12.0.922417564483584.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$18$ | 18.6.19830463701420724967320173.1 | \(\Z/9\Z\) | not in database |
$18$ | 18.0.1598763077775622674562101105291264.3 | \(\Z/9\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | ord | ord | ord | ord | add | ord | ord | ss | ord | ord | ord | ss | ord | ord |
$\lambda$-invariant(s) | 2 | 2 | 0 | 2 | 0 | - | 0 | 0 | 0,0 | 0 | 0 | 0 | 0,0 | 0 | 0 |
$\mu$-invariant(s) | 0 | 1 | 0 | 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.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.