Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-86x-2456\)
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(homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-86xz^2-2456z^3\)
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(dehomogenize, simplify) |
\(y^2=x^3-110835x-114242994\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 38 \) | = | $2 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-2550136832 $ | = | $-1 \cdot 2^{27} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{69173457625}{2550136832} \) | = | $-1 \cdot 2^{-27} \cdot 5^{3} \cdot 19^{-1} \cdot 821^{3}$ | 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: | $0.48516961095724358095196192008\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.48516961095724358095196192008\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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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: | $0.63021074331409951207066938986\dots$ | 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: | $ 1 $ | 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) $ ≈ $ 0.63021074331409951207066938986 $ | 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
$\displaystyle 0.630210743 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.630211 \cdot 1.000000 \cdot 1}{1^2} \approx 0.630210743$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 18 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 2 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{27}$ | Non-split multiplicative | 1 | 1 | 27 | 27 |
$19$ | $1$ | $I_{1}$ | 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.1.2 | 27.72.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 4104 = 2^{3} \cdot 3^{3} \cdot 19 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 28 & 27 \\ 1521 & 3592 \end{array}\right),\left(\begin{array}{rr} 937 & 75 \\ 1653 & 2144 \end{array}\right),\left(\begin{array}{rr} 31 & 36 \\ 2386 & 1447 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 688 & 9 \\ 791 & 664 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 3079 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4051 & 54 \\ 4050 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[4104])$ is a degree-$45954293760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/4104\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 38.a
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | 2.0.3.1-1444.2-b4 |
$3$ | 3.1.152.1 | \(\Z/2\Z\) | Not in database |
$3$ | 3.1.1083.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.3511808.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$6$ | 6.0.2565108243.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.7105563.1 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.623808.1 | \(\Z/6\Z\) | Not in database |
$9$ | 9.1.12356882919936.1 | \(\Z/6\Z\) | Not in database |
$12$ | 12.2.119973433931988992.10 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.8990607867641856.3 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.16877848680315122776257224907.3 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$18$ | 18.0.4122698998419163225428590592.1 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.1597218061967362443766361801733439488.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.12256029818428054141438155030528.1 | \(\Z/18\Z\) | Not in database |
$18$ | 18.0.1485393179874874809517382565888.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
$p$ | 2 | 3 | 19 |
---|---|---|---|
Reduction type | nonsplit | ord | split |
$\lambda$-invariant(s) | 1 | 0 | 1 |
$\mu$-invariant(s) | 0 | 2 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.