Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-18725758x+31284762394\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-18725758xz^2+31284762394z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-24268582800x+1459913097258000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2207, 26680)$ | $0.39104710520184050807155317343$ | $\infty$ |
| $(4748, 222337)$ | $0.41672451756626471169244466641$ | $\infty$ |
Integral points
\( \left(-4327, 176962\right) \), \( \left(-4327, -176963\right) \), \( \left(-3722, 222337\right) \), \( \left(-3722, -222338\right) \), \( \left(-1726, 241798\right) \), \( \left(-1726, -241799\right) \), \( \left(-1027, 222337\right) \), \( \left(-1027, -222338\right) \), \( \left(-397, 196612\right) \), \( \left(-397, -196613\right) \), \( \left(1514, 80041\right) \), \( \left(1514, -80042\right) \), \( \left(2207, 26680\right) \), \( \left(2207, -26681\right) \), \( \left(2447, 10999\right) \), \( \left(2447, -11000\right) \), \( \left(2603, 13612\right) \), \( \left(2603, -13613\right) \), \( \left(2648, 16537\right) \), \( \left(2648, -16538\right) \), \( \left(3278, 71662\right) \), \( \left(3278, -71663\right) \), \( \left(4748, 222337\right) \), \( \left(4748, -222338\right) \), \( \left(13673, 1526962\right) \), \( \left(13673, -1526963\right) \), \( \left(19994, 2765878\right) \), \( \left(19994, -2765879\right) \), \( \left(31973, 5667337\right) \), \( \left(31973, -5667338\right) \), \( \left(443648, 295486537\right) \), \( \left(443648, -295486538\right) \)
Invariants
| Conductor: | $N$ | = | \( 444675 \) | = | $3 \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $\Delta$ | = | $-2742040655047461796875$ | = | $-1 \cdot 3^{7} \cdot 5^{7} \cdot 7^{7} \cdot 11^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | $j$ | = | \( -\frac{222985990144}{841995} \) | = | $-1 \cdot 2^{12} \cdot 3^{-7} \cdot 5^{-1} \cdot 7^{-1} \cdot 11^{-1} \cdot 379^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9729903138975793473472962220$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.0036313532463127645367316053160$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $Q$ | ≈ | $0.8908239351519434$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.75631432894903$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Mordell-Weil rank: | $r$ | = | $ 2$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.15734750688516797785168778042$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $\Omega$ | ≈ | $0.14423208227826175698057247894$ | 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: | $\prod_{p}c_p$ | = | $ 448 $ = $ 7\cdot2^{2}\cdot2^{2}\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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ | 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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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.167162234584725927270955376 $ | 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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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 10.167162235 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.144232 \cdot 0.157348 \cdot 448}{1^2} \approx 10.167162235$
Modular invariants
Modular form 444675.2.a.bd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 46448640 | 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 at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $11$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1541 & 2 \\ 1541 & 3 \end{array}\right),\left(\begin{array}{rr} 211 & 2 \\ 211 & 3 \end{array}\right),\left(\begin{array}{rr} 661 & 2 \\ 661 & 3 \end{array}\right),\left(\begin{array}{rr} 1387 & 2 \\ 1387 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2309 & 2 \\ 2308 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 2309 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[2310])$ is a degree-$1839366144000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2310\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $3$ | split multiplicative | $4$ | \( 148225 = 5^{2} \cdot 7^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 17787 = 3 \cdot 7^{2} \cdot 11^{2} \) |
| $7$ | additive | $32$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 444675bd consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1155i1, its twist by $385$.
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.