Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-1358150x-661756089\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-1358150xz^2-661756089z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-21730395x-42374120074\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(\frac{5435}{4}, -\frac{5439}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 6498 \) | = | $2 \cdot 3^{2} \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-29043118367986164642 $ | = | $-1 \cdot 2 \cdot 3^{8} \cdot 19^{12} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{8078253774625}{846825858} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 5^{3} \cdot 19^{-6} \cdot 4013^{3}$ | 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);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.4736012701473504815360799644\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.45207563623007540583394363000\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0101536552434738\dots$ | |||
Szpiro ratio: | $6.167381996804376\dots$ |
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: | $0.069513019128923026209118759390\dots$ | 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: | $ 16 $ = $ 1\cdot2^{2}\cdot2^{2} $ | 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: | $2$ | 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: | $9$ = $3^2$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L(E,1) $ ≈ $ 2.5024686886412289435282753381 $ | 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
$\displaystyle 2.502468689 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.069513 \cdot 1.000000 \cdot 16}{2^2} \approx 2.502468689$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 207360 | 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 $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
$3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
$19$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
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 | 8.6.0.5 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 456 = 2^{3} \cdot 3 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 406 & 447 \end{array}\right),\left(\begin{array}{rr} 359 & 444 \\ 330 & 383 \end{array}\right),\left(\begin{array}{rr} 210 & 145 \\ 437 & 362 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 151 & 444 \\ 374 & 383 \end{array}\right),\left(\begin{array}{rr} 445 & 12 \\ 444 & 13 \end{array}\right),\left(\begin{array}{rr} 10 & 3 \\ 201 & 448 \end{array}\right)$.
The torsion field $K:=\Q(E[456])$ is a degree-$94556160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$.
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 | $4$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
$3$ | additive | $8$ | \( 722 = 2 \cdot 19^{2} \) |
$19$ | additive | $200$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 6498.t
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 114.c2, its twist by $57$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{-19}) \) | \(\Z/6\Z\) | not in database |
$4$ | 4.2.103968.1 | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$6$ | 6.2.19440820368.5 | \(\Z/6\Z\) | not in database |
$8$ | 8.0.691798081536.13 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.44275077218304.53 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.10809345024.5 | \(\Z/12\Z\) | not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$12$ | deg 12 | \(\Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/8\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$18$ | 18.0.800168814247321224269671488563725056.3 | \(\Z/18\Z\) | 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 | 19 |
---|---|---|---|
Reduction type | split | add | add |
$\lambda$-invariant(s) | 6 | - | - |
$\mu$-invariant(s) | 1 | - | - |
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 $0$.