Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-441x+3672\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-441xz^2+3672z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-7059x+227950\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(-24, 12)$ | $0$ | $2$ |
$(12, -6)$ | $0$ | $2$ |
Integral points
\( \left(-24, 12\right) \), \( \left(12, -6\right) \)
Invariants
Conductor: | $N$ | = | \( 63 \) | = | $3^{2} \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $\Delta$ | = | $15752961$ | = | $3^{8} \cdot 7^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | $j$ | = | \( \frac{13027640977}{21609} \) | = | $3^{-2} \cdot 7^{-4} \cdot 13^{3} \cdot 181^{3}$ | 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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.27693793744502196174727241241$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.27236820688903288395035020605$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $Q$ | ≈ | $1.0814902740568197$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.212416412105554$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Mordell-Weil rank: | $r$ | = | $ 0$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $\Omega$ | ≈ | $2.2066209286713850140325004382$ | 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: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2 $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ | 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(E,1)$ | ≈ | $1.1033104643356925070162502191 $ | 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${}_{\mathrm{an}}$ | = | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 1.103310464 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 2.206621 \cdot 1.000000 \cdot 8}{4^2} \approx 1.103310464$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 16 | 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 at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
$7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
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 |
---|---|---|
$2$ | 2Cs | 8.48.0.18 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 89 & 44 \\ 64 & 93 \end{array}\right),\left(\begin{array}{rr} 73 & 8 \\ 124 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 164 & 165 \end{array}\right),\left(\begin{array}{rr} 7 & 48 \\ 162 & 121 \end{array}\right),\left(\begin{array}{rr} 161 & 8 \\ 160 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 49 & 166 \\ 18 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$774144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 |
---|---|---|---|
$2$ | good | $2$ | \( 9 = 3^{2} \) |
$3$ | additive | $8$ | \( 7 \) |
$7$ | nonsplit multiplicative | $8$ | \( 9 = 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 63.a
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 21.a2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.2.12.1-147.1-b6 |
$2$ | \(\Q(\sqrt{-3}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.3.1-147.2-a7 |
$4$ | \(\Q(\zeta_{12})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
$4$ | 4.0.432.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.8.12745506816.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.12745506816.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.0.2985984.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.2.425329947.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.162447943996702457856.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
$16$ | 16.0.49323011734109159424.1 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 7 |
---|---|---|---|
Reduction type | ord | add | nonsplit |
$\lambda$-invariant(s) | 2 | - | 0 |
$\mu$-invariant(s) | 0 | - | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
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 $0$.