Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-46938x-3991412\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-46938xz^2-3991412z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-60831027x-186040813554\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{1051}{4}, \frac{9775}{8}\right)\) |
$\hat{h}(P)$ | ≈ | $3.3310156590204861156880745335$ |
Integral points
None
Invariants
Conductor: | \( 400710 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2} \cdot 37$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-250660453968000 $ | = | $-1 \cdot 2^{7} \cdot 3^{2} \cdot 5^{3} \cdot 19^{6} \cdot 37 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{243087455521}{5328000} \) | = | $-1 \cdot 2^{-7} \cdot 3^{-2} \cdot 5^{-3} \cdot 37^{-1} \cdot 79^{6}$ | 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: | $1.5521151099418415091267489313\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.079895620358621279122235215357\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.071214811402831\dots$ | |||
Szpiro ratio: | $3.404427463481029\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $3.3310156590204861156880745335\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.16196685597648741724636531258\dots$ | 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: | $ 12 $ = $ 1\cdot2\cdot3\cdot2\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) $ ≈ $ 6.4741696019999447329593305332 $ | 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
$\displaystyle 6.474169602 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.161967 \cdot 3.331016 \cdot 12}{1^2} \approx 6.474169602$
Modular invariants
Modular form 400710.2.a.s
For more coefficients, see the Downloads section to the right.
Modular degree: | 2201472 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
$3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$19$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$37$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1480 = 2^{3} \cdot 5 \cdot 37 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1479 & 2 \\ 1478 & 3 \end{array}\right),\left(\begin{array}{rr} 1111 & 2 \\ 1111 & 3 \end{array}\right),\left(\begin{array}{rr} 741 & 2 \\ 741 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 1479 & 0 \end{array}\right),\left(\begin{array}{rr} 297 & 2 \\ 297 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1001 & 2 \\ 1001 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[1480])$ is a degree-$671726960640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1480\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 400710s consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1110j1, its twist by $-19$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.