Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-1080x-13662\)
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(homogenize, simplify) |
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\(y^2z=x^3-1080xz^2-13662z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1080x-13662\)
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(homogenize, minimize) |
comment: Define the curve
sage: E = EllipticCurve([0, 0, 0, -1080, -13662])
gp: E = ellinit([0, 0, 0, -1080, -13662])
magma: E := EllipticCurve([0, 0, 0, -1080, -13662]);
oscar: E = elliptic_curve([0, 0, 0, -1080, -13662])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Mordell-Weil group structure
\(\Z\)
magma: MordellWeilGroup(E);
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(871, 25687\right)\)
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| $\hat{h}(P)$ | ≈ | $6.7911921286310453797986406061$ |
sage: E.gens()
magma: Generators(E);
gp: E.gen
Integral points
\((871,\pm 25687)\)
comment: Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
| Conductor: | \( 1728 \) | = | $2^{6} \cdot 3^{3}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-11337408 $ | = | $-1 \cdot 2^{6} \cdot 3^{11} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -12288000 \) | = | $-1 \cdot 2^{15} \cdot 3 \cdot 5^{3}$ | 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: | \(\Z[(1+\sqrt{-27})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $0.39872152268707185396280341103\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-0.95491333220533468452478745021\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $1.23864473399791\dots$ | |||
| Szpiro ratio: | $4.36876137928833\dots$ | |||
BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $6.7911921286310453797986406061\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.41640074674458981194300751834\dots$ | 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: | $ 1 $ | 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) $ ≈ $ 2.8278574736479477248342302820 $ | 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
$\displaystyle 2.827857474 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.416401 \cdot 6.791192 \cdot 1}{1^2} \approx 2.827857474$
# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
Modular invariants
\( q - q^{7} - 5 q^{13} + 7 q^{19} + O(q^{20}) \)
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comment: q-expansion of modular form
sage: E.q_eigenform(20)
\\ actual modular form, use for small N
[mf,F] = mffromell(E)
Ser(mfcoefs(mf,20),q)
\\ or just the series
Ser(ellan(E,20),q)*q
magma: ModularForm(E);
For more coefficients, see the Downloads section to the right.
| Modular degree: | 432 | 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
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $v_p(N)$ | $v_p(\Delta)$ | $v_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II$ | additive | 1 | 6 | 6 | 0 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 3 | 11 | 0 |
comment: Local data
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
comment: mod p Galois image
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 27 = 3^{3} \) |
| $3$ | additive | $4$ | \( 64 = 2^{6} \) |
Isogenies
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 9 and 27.
Its isogeny class 1728a
consists of 4 curves linked by isogenies of
degrees dividing 27.
Twists
The minimal quadratic twist of this elliptic curve is 27a4, its twist by $-24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.108.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.34992.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.30233088.6 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.10077696.1 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.4478976.2 | \(\Z/6\Z\) | not in database |
| $12$ | 12.2.962938848411648.4 | \(\Z/4\Z\) | not in database |
| $12$ | 12.0.8226356490141696.32 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | 12.0.8226356490141696.34 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | not in database |
| $12$ | 12.0.20061226008576.9 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.396521139274783615537700143104.1 | \(\Z/27\Z\) | not in database |
| $18$ | 18.2.1289303099811316943271493632.4 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.5305774073297600589594624.1 | \(\Z/18\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ss | ord | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | - | - | 1,1 | 1 | 1,1 | 1 | 1,1 | 1 | 1,1 | 1,1 | 1 | 1 | 1,1 | 1 | 1,1 |
| $\mu$-invariant(s) | - | - | 0,0 | 0 | 0,0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.