LMFDB - Elliptic curve with Cremona label 430950cz1 (LMFDB label 430950.cz1)

Show commands: Magma / Oscar / PariGP / SageMath

Simplified equation

$y^2+xy+y=x^3-2526x-632552$ y^2+xy+y=x^3-2526x-632552	(homogenize, simplify)
$y^2z+xyz+yz^2=x^3-2526xz^2-632552z^3$ y^2z+xyz+yz^2=x^3-2526xz^2-632552z^3	(dehomogenize, simplify)
$y^2=x^3-3273075x-29502515250$ y^2=x^3-3273075x-29502515250	(homogenize, minimize)

comment: Define the curve

sage: E = EllipticCurve([1, 0, 1, -2526, -632552])

gp: E = ellinit([1, 0, 1, -2526, -632552])

magma: E := EllipticCurve([1, 0, 1, -2526, -632552]);

oscar: E = EllipticCurve([1, 0, 1, -2526, -632552])

sage: E.short_weierstrass_model()

magma: WeierstrassModel(E);

oscar: short_weierstrass_model(E)

Mordell-Weil group structure

$\Z \oplus \Z$

magma: MordellWeilGroup(E);

Infinite order Mordell-Weil generators and heights

$P$	=	$\left(122, 876\right)$ (122, 876)	$\left(158, 1629\right)$ (158, 1629)
$\hat{h}(P)$	≈	$1.0401143647899682434257755332$	$3.4194487135265430025126892901$

sage: E.gens()

magma: Generators(E);

gp: E.gen

$ \left(122, 876\right) $, $ \left(122, -999\right) $, $ \left(158, 1629\right) $, $ \left(158, -1788\right) $, $ \left(262, 3956\right) $, $ \left(262, -4219\right) $, $ \left(5122, 364001\right) $, $ \left(5122, -369124\right) $ (122, 876), (122, -999), (158, 1629), (158, -1788), (262, 3956), (262, -4219), (5122, 364001), (5122, -369124)

comment: Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

Invariants

Conductor:	$ 430950 $	=	$2 \cdot 3 \cdot 5^{2} \cdot 13^{2} \cdot 17$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$-171706640625000 $	=	$-1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{11} \cdot 13^{2} \cdot 17^{2} $	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	$ -\frac{674636521}{65025000} $	=	$-1 \cdot 2^{-3} \cdot 3^{-2} \cdot 5^{-5} \cdot 13 \cdot 17^{-2} \cdot 373^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	$\Z$	(no potential complex multiplication)		sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$1.4109650306288666278342170714\dots$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$0.17875451483489365119158949786\dots$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$0.9272967709473774\dots$
Szpiro ratio:	$3.101456620863791\dots$

BSD invariants

Analytic rank:	$2$	sage: E.analytic_rank() gp: ellanalyticrank(E) magma: AnalyticRank(E);
Regulator:	$3.3331830275184119922379224413\dots$	comment: Regulator sage: E.regulator() G = E.gen \\ if available matdet(ellheightmatrix(E,G)) magma: Regulator(E);
Real period:	$0.25303110833619524035055915034\dots$	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);
Tamagawa product:	$ 16 $ = $ 1\cdot2\cdot2^{2}\cdot1\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)
Torsion order:	$1$	comment: Torsion order sage: E.torsion_order() gp: elltors(E)[1] magma: Order(TorsionSubgroup(E)); oscar: prod(torsion_structure(E)[1])
Analytic order of Ш:	$1$ ( rounded)	comment: Order of Sha sage: E.sha().an_numerical() magma: MordellWeilShaInformation(E);
Special value:	$ L^{(2)}(E,1)/2! $ ≈ $ 13.494383931846056733846493586 $	comment: Special L-value r = E.rank(); E.lseries().dokchitser().derivative(1,r)/r.factorial() gp: [r,L1r] = ellanalyticrank(E); L1r/r! magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);

BSD formula

$\displaystyle 13.494383932 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.253031 \cdot 3.333183 \cdot 16}{1^2} \approx 13.494383932$

# 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 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

Modular form 430950.2.a.cz

$ q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} - 5 q^{11} + q^{12} + q^{14} + q^{16} - q^{17} - q^{18} + q^{19} + O(q^{20}) $

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:	1520640	comment: Modular degree sage: E.modular_degree() gp: ellmoddegree(E) magma: ModularDegree(E);
$ \Gamma_0(N) $-optimal:	yes
Manin constant:	1	comment: Manin constant magma: ManinConstant(E);

Local data

This elliptic curve is not semistable. There are 5 primes of bad reduction:

prime	Tamagawa number	Kodaira symbol	Reduction type	Root number	ord($N$)	ord($\Delta$)	ord$(j)_{-}$
$2$	$1$	$I_{3}$	Non-split multiplicative	1	1	3	3
$3$	$2$	$I_{2}$	Split multiplicative	-1	1	2	2
$5$	$4$	$I_{5}^{*}$	Additive	1	2	11	5
$13$	$1$	$II$	Additive	1	2	2	0
$17$	$2$	$I_{2}$	Non-split multiplicative	1	1	2	2

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)];

gens = [[21, 2, 21, 3], [1, 0, 2, 1], [17, 2, 17, 3], [31, 2, 31, 3], [1, 2, 0, 1], [1, 1, 39, 0], [39, 2, 38, 3]]

GL(2,Integers(40)).subgroup(gens)

Gens := [[21, 2, 21, 3], [1, 0, 2, 1], [17, 2, 17, 3], [31, 2, 31, 3], [1, 2, 0, 1], [1, 1, 39, 0], [39, 2, 38, 3]];

sub<GL(2,Integers(40))|Gens>;

The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 40.2.0.a.1, level $ 40 = 2^{3} \cdot 5 $, index $2$, genus $0$, and generators

$\left(\begin{array}{rr} 21 & 2 \\ 21 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 17 & 2 \\ 17 & 3 \end{array}\right),\left(\begin{array}{rr} 31 & 2 \\ 31 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 39 & 0 \end{array}\right),\left(\begin{array}{rr} 39 & 2 \\ 38 & 3 \end{array}\right)$.

The torsion field $K:=\Q(E[40])$ is a degree-$368640$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/40\Z)$.

$p$-adic regulators

$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.

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Learn more

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

Mordell-Weil group structure

Infinite order Mordell-Weil generators and heights

Integral points

Invariants

BSD invariants

BSD formula

Modular invariants

Local data

Galois representations

Isogenies

Twists

Iwasawa invariants

$p$-adic regulators

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	430950cz1

Conductor	$430950$
Discriminant	$-1.717\times 10^{14}$
j-invariant	\( -\frac{674636521}{65025000} \)
CM	no
Rank	$2$
Torsion structure	trivial

\(y^2+xy+y=x^3-2526x-632552\) y^2+xy+y=x^3-2526x-632552	(homogenize, simplify)
\(y^2z+xyz+yz^2=x^3-2526xz^2-632552z^3\) y^2z+xyz+yz^2=x^3-2526xz^2-632552z^3	(dehomogenize, simplify)
\(y^2=x^3-3273075x-29502515250\) y^2=x^3-3273075x-29502515250	(homogenize, minimize)

$P$	=	\(\left(122, 876\right)\) (122, 876)	\(\left(158, 1629\right)\) (158, 1629)
$\hat{h}(P)$	≈	$1.0401143647899682434257755332$	$3.4194487135265430025126892901$

Conductor:	\( 430950 \)	=	$2 \cdot 3 \cdot 5^{2} \cdot 13^{2} \cdot 17$	comment: Conductor sage: E.conductor().factor() gp: ellglobalred(E)[1] magma: Conductor(E); oscar: conductor(E)
Discriminant:	$-171706640625000 $	=	$-1 \cdot 2^{3} \cdot 3^{2} \cdot 5^{11} \cdot 13^{2} \cdot 17^{2} $	comment: Discriminant sage: E.discriminant().factor() gp: E.disc magma: Discriminant(E); oscar: discriminant(E)
j-invariant:	\( -\frac{674636521}{65025000} \)	=	$-1 \cdot 2^{-3} \cdot 3^{-2} \cdot 5^{-5} \cdot 13 \cdot 17^{-2} \cdot 373^{3}$	comment: j-invariant sage: E.j_invariant().factor() gp: E.j magma: jInvariant(E); oscar: j_invariant(E)
Endomorphism ring:	$\Z$
Geometric endomorphism ring:	\(\Z\)	(no potential complex multiplication)		sage: E.has_cm() magma: HasComplexMultiplication(E);
Sato-Tate group:	$\mathrm{SU}(2)$
Faltings height:	$1.4109650306288666278342170714\dots$			gp: ellheight(E) magma: FaltingsHeight(E); oscar: faltings_height(E)
Stable Faltings height:	$0.17875451483489365119158949786\dots$			magma: StableFaltingsHeight(E); oscar: stable_faltings_height(E)
$abc$ quality:	$0.9272967709473774\dots$
Szpiro ratio:	$3.101456620863791\dots$