Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+1\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+80\)
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(homogenize, minimize) |
comment: Define the curve
sage: E = EllipticCurve([0, 0, 1, 0, 1])
gp: E = ellinit([0, 0, 1, 0, 1])
magma: E := EllipticCurve([0, 0, 1, 0, 1]);
oscar: E = EllipticCurve([0, 0, 1, 0, 1])
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
| $P$ | = |
\(\left(1, 1\right)\)
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| $\hat{h}(P)$ | ≈ | $0.15364003501329447809907699066$ |
sage: E.gens()
magma: Generators(E);
gp: E.gen
Integral points
\( \left(-1, 0\right) \), \( \left(-1, -1\right) \), \( \left(1, 1\right) \), \( \left(1, -2\right) \), \( \left(11, 36\right) \), \( \left(11, -37\right) \)
comment: Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
Invariants
| Conductor: | \( 225 \) | = | $3^{2} \cdot 5^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-675 $ | = | $-1 \cdot 3^{3} \cdot 5^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( 0 \) | = | $0$ | 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: | \(\Z[(1+\sqrt{-3})/2]\) | (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
| Sato-Tate group: | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $-0.77822470418866042970759799775\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $-1.3211174284280379149898691959\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.15364003501329447809907699066\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $4.0529757590368917612716776534\dots$ | 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: | $ 2 $ = $ 2\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: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 1.2453986750529236279151961986 $ | 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
$\displaystyle 1.245398675 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 4.052976 \cdot 0.153640 \cdot 2}{1^2} \approx 1.245398675$
# 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("anayltic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analyiic 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
\( q - 2 q^{4} - 5 q^{7} - 5 q^{13} + 4 q^{16} - q^{19} + O(q^{20}) \)
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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: | 8 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
| $5$ | $1$ | $II$ | Additive | 1 | 2 | 2 | 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 $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $5$ | 5Ns.2.1 | 5.30.0.2 |
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)];
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ss | add | add | ord | ss | ord | ss | ord | ss | ss | ord | ord | ss | ord | ss |
| $\lambda$-invariant(s) | ? | - | - | 1 | 1,3 | 1 | 1,1 | 1 | 1,3 | 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 |
An entry ? indicates that the invariants have not yet been computed.
An entry - indicates that the invariants are not computed because the reduction is additive.
Isogenies
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 225.c
consists of 2 curves linked by isogenies of
degree 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/3\Z\) | 2.2.5.1-2025.1-d1 |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $6$ | 6.0.6834375.1 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.2.450000.1 | \(\Z/6\Z\) | Not in database |
| $8$ | 8.4.56953125.1 | \(\Z/15\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | 12.0.46708681640625.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | Not in database |
| $12$ | 12.0.1822500000000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | 16.0.3243658447265625.1 | \(\Z/5\Z \oplus \Z/15\Z\) | Not in database |
| $18$ | 18.6.6283298708943145751953125.4 | \(\Z/9\Z\) | Not in database |
| $18$ | 18.0.1307544150375000000000000.1 | \(\Z/6\Z\) | 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.