Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+262412587x+7228069259906\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+262412587xz^2+7228069259906z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+340086712725x+337227698089491750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(-13855, 972827\right)\)
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| $\hat{h}(P)$ | ≈ | $8.7131687762744233308555259695$ |
Torsion generators
\( \left(-\frac{59645}{4}, \frac{59641}{8}\right) \)
Integral points
\( \left(-13855, 972827\right) \), \( \left(-13855, -958973\right) \)
Invariants
| Conductor: | \( 441525 \) | = | $3 \cdot 5^{2} \cdot 7 \cdot 29^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-23725619708509447845992859375 $ | = | $-1 \cdot 3^{6} \cdot 5^{6} \cdot 7 \cdot 29^{14} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( \frac{215015459663151503}{2552757445339983} \) | = | $3^{-6} \cdot 7^{-1} \cdot 29^{-8} \cdot 599087^{3}$ | 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: | $4.1256022447222017898576291985\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $1.6372353735119145889656135157\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.0258598043382023\dots$ | |||
| Szpiro ratio: | $5.5963346531588165\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $8.7131687762744233308555259695\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.027996324380607619108746051955\dots$ | 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 $ = $ 2\cdot2\cdot1\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: | $4$ = $2^2$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 3.9029871910969710610282425795 $ | 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
$\displaystyle 3.902987191 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{4 \cdot 0.027996 \cdot 8.713169 \cdot 16}{2^2} \approx 3.902987191$
Modular invariants
Modular form 441525.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 330301440 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
| $29$ | $4$ | $I_{8}^{*}$ | Additive | 1 | 2 | 14 | 8 |
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 | 8.12.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 48720 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 29 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 48705 & 16 \\ 48704 & 17 \end{array}\right),\left(\begin{array}{rr} 9743 & 0 \\ 0 & 48719 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 48716 & 48717 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 48622 & 48707 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 26456 & 9745 \\ 4255 & 38986 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 35741 & 9760 \\ 22480 & 38661 \end{array}\right),\left(\begin{array}{rr} 19501 & 9760 \\ 2180 & 2121 \end{array}\right),\left(\begin{array}{rr} 25591 & 9760 \\ 8270 & 2121 \end{array}\right),\left(\begin{array}{rr} 25199 & 38960 \\ 26200 & 19359 \end{array}\right)$.
The torsion field $K:=\Q(E[48720])$ is a degree-$4055256111513600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/48720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 441525k
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 609b6, its twist by $145$.
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.