Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-280713x+57615820\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-280713xz^2+57615820z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-363803427x+2689215119838\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(251, 1602\right)\) |
$\hat{h}(P)$ | ≈ | $1.1901347148692661762805027598$ |
Integral points
\( \left(200, 2979\right) \), \( \left(200, -3180\right) \), \( \left(251, 1602\right) \), \( \left(251, -1854\right) \)
Invariants
Conductor: | \( 114582 \) | = | $2 \cdot 3 \cdot 13^{2} \cdot 113$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-19543722147495936 $ | = | $-1 \cdot 2^{14} \cdot 3^{7} \cdot 13^{6} \cdot 113 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{506814405937489}{4048994304} \) | = | $-1 \cdot 2^{-14} \cdot 3^{-7} \cdot 13^{3} \cdot 113^{-1} \cdot 6133^{3}$ | 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: | $1.9539668813719898925280889212\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.67149220264122152450134520042\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9749842201837565\dots$ | |||
Szpiro ratio: | $4.228893325227882\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.1901347148692661762805027598\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.38747077067974596938672842643\dots$ | 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);
|
Tamagawa product: | $ 14 $ = $ 2\cdot7\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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 6.4559938125636000616515675563 $ | 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
$\displaystyle 6.455993813 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.387471 \cdot 1.190135 \cdot 14}{1^2} \approx 6.455993813$
Modular invariants
Modular form 114582.2.a.h
For more coefficients, see the Downloads section to the right.
Modular degree: | 889056 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{14}$ | Non-split multiplicative | 1 | 1 | 14 | 14 |
$3$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
$13$ | $1$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
$113$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$7$ | 7B.6.1 | 7.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 61698 = 2 \cdot 3 \cdot 7 \cdot 13 \cdot 113 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 61685 & 14 \\ 61684 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 56966 \\ 0 & 17629 \end{array}\right),\left(\begin{array}{rr} 56951 & 0 \\ 0 & 61697 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 55147 & 56966 \\ 44317 & 28575 \end{array}\right),\left(\begin{array}{rr} 20567 & 56966 \\ 49049 & 28575 \end{array}\right)$.
The torsion field $K:=\Q(E[61698])$ is a degree-$25613268533379072$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/61698\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 114582f
consists of 2 curves linked by isogenies of
degree 7.
Twists
The minimal quadratic twist of this elliptic curve is 678d1, 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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{13}) \) | \(\Z/7\Z\) | Not in database |
$3$ | 3.1.339.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.38958219.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.252481437.1 | \(\Z/14\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/14\Z\) | Not in database |
$16$ | deg 16 | \(\Z/21\Z\) | 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 | 113 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | nonsplit | split | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ss | ord | ord | split |
$\lambda$-invariant(s) | 1 | 2 | 1 | 1 | 1 | - | 3 | 3 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 2 |
$\mu$-invariant(s) | 0 | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.