Minimal Weierstrass equation
sage: E = EllipticCurve([0, 0, 0, -152955, 23024682])
gp: E = ellinit([0, 0, 0, -152955, 23024682])
magma: E := EllipticCurve([0, 0, 0, -152955, 23024682]);
Minimal equation
Minimal equation
Simplified equation
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\(y^2=x^3-152955x+23024682\)
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(homogenize, simplify) |
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\(y^2z=x^3-152955xz^2+23024682z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-152955x+23024682\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 3024 \) | = | $2^{4} \cdot 3^{3} \cdot 7$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-10158317568 $ | = | $-1 \cdot 2^{13} \cdot 3^{11} \cdot 7 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{545407363875}{14} \) | = | $-1 \cdot 2^{-1} \cdot 3 \cdot 5^{3} \cdot 7^{-1} \cdot 11^{3} \cdot 103^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $1.4363371825903282851839165368\dots$ | ||
| Stable Faltings height: | $-0.26387126258205090801229038517\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.93694636751196903439418010379\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 2 $ = $ 2\cdot1\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $1$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $1$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L(E,1) $ ≈ $ 1.8738927350239380687883602076 $ | ||
Modular invariants
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
magma: ModularForm(E);
\( q - q^{7} + 5 q^{13} + 3 q^{17} - 2 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 7776 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | Additive | -1 | 4 | 13 | 1 |
| $3$ | $1$ | $II^{*}$ | Additive | 1 | 3 | 11 | 0 |
| $7$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
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 |
|---|---|---|
| $3$ | 3B | 9.12.0.1 |
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
All $p$-adic regulators are identically $1$ since the rank is $0$.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ss | nonsplit | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 0,0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | - | - | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 3024.p
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/3\Z\) | Not in database |
| $3$ | 3.1.1512.1 | \(\Z/2\Z\) | Not in database |
| $6$ | 6.0.384072192.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $6$ | 6.0.36294822144.16 | \(\Z/3\Z\) | Not in database |
| $6$ | 6.6.4148928.1 | \(\Z/9\Z\) | Not in database |
| $6$ | 6.2.109734912.1 | \(\Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/4\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/9\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $18$ | 18.0.1748870220375544658742100451921951195136.1 | \(\Z/6\Z\) | Not in database |
| $18$ | 18.6.7617610788994000809356509052928.1 | \(\Z/18\Z\) | 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.