Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-21681x-6833646\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-21681xz^2-6833646z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-28098603x-318746291994\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 146523 \) | = | $3 \cdot 13^{2} \cdot 17^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-19510922280339009 $ | = | $-1 \cdot 3^{14} \cdot 13^{2} \cdot 17^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{276301129}{4782969} \) | = | $-1 \cdot 3^{-14} \cdot 13 \cdot 277^{3}$ | 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: | $1.8069749946356808807996532348\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.037123236969349948667361981065\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.06786710909304\dots$ | |||
Szpiro ratio: | $3.782899555584777\dots$ |
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: | $0.16574690375787935291401347676\dots$ | 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: | $ 14 $ = $ ( 2 \cdot 7 )\cdot1\cdot1 $ | 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) $ ≈ $ 2.3204566526103109407961886747 $ | 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
$\displaystyle 2.320456653 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.165747 \cdot 1.000000 \cdot 14}{1^2} \approx 2.320456653$
Modular invariants
Modular form 146523.2.a.n
For more coefficients, see the Downloads section to the right.
Modular degree: | 739200 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $14$ | $I_{14}$ | Split multiplicative | -1 | 1 | 14 | 14 |
$13$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 0 |
$17$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
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 |
---|---|---|
$2$ | 2G | 4.2.0.1 |
$7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 37128 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \cdot 17 \), index $192$, genus $6$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 28 & 1 \end{array}\right),\left(\begin{array}{rr} 9283 & 15300 \\ 28679 & 28561 \end{array}\right),\left(\begin{array}{rr} 31825 & 10948 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8735 & 0 \\ 0 & 37127 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 37101 & 28 \\ 37100 & 29 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 36883 & 36975 \end{array}\right),\left(\begin{array}{rr} 23189 & 10948 \\ 8806 & 7549 \end{array}\right),\left(\begin{array}{rr} 12377 & 15300 \\ 11662 & 28561 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 14 & 197 \end{array}\right),\left(\begin{array}{rr} 18565 & 15300 \\ 24038 & 28561 \end{array}\right)$.
The torsion field $K:=\Q(E[37128])$ is a degree-$1589340769615872$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/37128\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 146523.n
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 507.b1, its twist by $17$.
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 |
---|---|---|---|
$3$ | 3.1.676.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.1827904.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.30658699288763.4 | \(\Z/7\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$18$ | 18.0.118037805084712291186100215340993114976038912.1 | \(\Z/14\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | ord | split | ord | ord | ord | add | add | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 7 | 1 | 0 | 0 | 0 | - | - | 0 | 0 | 4 | 0 | 2 | 0 | 0 | 0 |
$\mu$-invariant(s) | 0 | 0 | 0 | 1 | 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 $0$.