Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-14597x-684237\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-14597xz^2-684237z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-1182384x-495261648\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 2352 \) | = | $2^{4} \cdot 3 \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-319987003392 $ | = | $-1 \cdot 2^{12} \cdot 3^{13} \cdot 7^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1713910976512}{1594323} \) | = | $-1 \cdot 2^{12} \cdot 3^{-13} \cdot 7 \cdot 17^{3} \cdot 23^{3}$ | 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: | $1.1305343428716613754189906903\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.11306880413583051515086644493\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1059237401535138\dots$ | |||
Szpiro ratio: | $5.201710776200118\dots$ |
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: | $0.21715855964856781142632468526\dots$ | 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: | $ 13 $ = $ 1\cdot13\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) $ ≈ $ 2.8230612754313815485422209084 $ | 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
$\displaystyle 2.823061275 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.217159 \cdot 1.000000 \cdot 13}{1^2} \approx 2.823061275$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 3120 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $II^{*}$ | Additive | -1 | 4 | 12 | 0 |
$3$ | $13$ | $I_{13}$ | Split multiplicative | -1 | 1 | 13 | 13 |
$7$ | $1$ | $II$ | Additive | -1 | 2 | 2 | 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 |
---|---|---|
$13$ | 13B.4.2 | 13.28.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $336$, genus $9$, and generators
$\left(\begin{array}{rr} 15 & 26 \\ 130 & 287 \end{array}\right),\left(\begin{array}{rr} 14 & 23 \\ 871 & 339 \end{array}\right),\left(\begin{array}{rr} 1067 & 26 \\ 1066 & 27 \end{array}\right),\left(\begin{array}{rr} 545 & 0 \\ 0 & 1091 \end{array}\right),\left(\begin{array}{rr} 1 & 26 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 571 & 1066 \\ 572 & 1065 \end{array}\right),\left(\begin{array}{rr} 1078 & 1069 \\ 767 & 753 \end{array}\right),\left(\begin{array}{rr} 485 & 26 \\ 156 & 101 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 26 & 1 \end{array}\right),\left(\begin{array}{rr} 365 & 26 \\ 377 & 339 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$724598784$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
13.
Its isogeny class 2352u
consists of 2 curves linked by isogenies of
degree 13.
Twists
The minimal quadratic twist of this elliptic curve is 147c2, its twist by $-4$.
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 |
---|---|---|---|
$3$ | 3.1.588.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.1037232.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.5335338855168.5 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.2073562945755298571112448.2 | \(\Z/13\Z\) | 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 | add | split | ord | add | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | 1 | 0 | - | 0 | 0 | 0,0 | 2 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
$\mu$-invariant(s) | - | 0 | 0 | - | 0 | 1 | 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 $0$.