Properties

 Label 369600hd4 Conductor $369600$ Discriminant $2.787\times 10^{23}$ j-invariant $$\frac{260744057755293612689909}{8504954620259328}$$ CM no Rank $1$ Torsion structure $$\Z/{2}\Z$$

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([0, -1, 0, -425905793, 3383174665857])

gp: E = ellinit([0, -1, 0, -425905793, 3383174665857])

magma: E := EllipticCurve([0, -1, 0, -425905793, 3383174665857]);

$$y^2=x^3-x^2-425905793x+3383174665857$$

Mordell-Weil group structure

$\Z\times \Z/{2}\Z$

Infinite order Mordell-Weil generator and height

sage: E.gens()

magma: Generators(E);

 $P$ = $$\left(5793, 1053696\right)$$ $\hat{h}(P)$ ≈ $1.3638063886790711191956465070$

Torsion generators

sage: E.torsion_subgroup().gens()

gp: elltors(E)

magma: TorsionSubgroup(E);

$$\left(11967, 0\right)$$

Integral points

sage: E.integral_points()

magma: IntegralPoints(E);

$$(5793,\pm 1053696)$$, $$\left(11967, 0\right)$$, $$(25827,\pm 3100020)$$

Invariants

 sage: E.conductor().factor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor: $$369600$$ = $2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$ sage: E.discriminant().factor()  gp: E.disc  magma: Discriminant(E); Discriminant: $278690352996657659904000$ = $2^{28} \cdot 3^{5} \cdot 5^{3} \cdot 7^{10} \cdot 11^{2}$ sage: E.j_invariant().factor()  gp: E.j  magma: jInvariant(E); j-invariant: $$\frac{260744057755293612689909}{8504954620259328}$$ = $2^{-10} \cdot 3^{-5} \cdot 7^{-10} \cdot 11^{-2} \cdot 29^{3} \cdot 2202961^{3}$ Endomorphism ring: $\Z$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) Sato-Tate group: $\mathrm{SU}(2)$ Faltings height: $3.5927228956905917617410426477\dots$ Stable Faltings height: $2.1506426467421487039650046322\dots$

BSD invariants

 sage: E.rank()  magma: Rank(E); Analytic rank: $1$ sage: E.regulator()  magma: Regulator(E); Regulator: $1.3638063886790711191956465070\dots$ sage: E.period_lattice().omega()  gp: E.omega[1]  magma: RealPeriod(E); Real period: $0.091177675570019157113520996913\dots$ sage: E.tamagawa_numbers()  gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]  magma: TamagawaNumbers(E); Tamagawa product: $160$  = $2^{2}\cdot1\cdot2\cdot( 2 \cdot 5 )\cdot2$ sage: E.torsion_order()  gp: elltors(E)[1]  magma: Order(TorsionSubgroup(E)); Torsion order: $2$ sage: E.sha().an_numerical()  magma: MordellWeilShaInformation(E); Analytic order of Ш: $1$ (exact) sage: r = E.rank(); sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()  gp: ar = ellanalyticrank(E); gp: ar[2]/factorial(ar[1])  magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12); Special value: $L'(E,1)$ ≈ $4.9739478578919917579514416949642869630$

Modular invariants

Modular form 369600.2.a.hd

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);

$$q - q^{3} + q^{7} + q^{9} - q^{11} - 4 q^{13} - 2 q^{17} + O(q^{20})$$

For more coefficients, see the Downloads section to the right.

 sage: E.modular_degree()  magma: ModularDegree(E); Modular degree: 73728000 $\Gamma_0(N)$-optimal: no Manin constant: 1

Local data

This elliptic curve is not semistable. There are 5 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$(j)_{-}$
$2$ $4$ $I_{18}^{*}$ Additive 1 6 28 10
$3$ $1$ $I_{5}$ Non-split multiplicative 1 1 5 5
$5$ $2$ $III$ Additive -1 2 3 0
$7$ $10$ $I_{10}$ Split multiplicative -1 1 10 10
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

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
$2$ 2B 2.3.0.1
$5$ 5B.4.1 5.12.0.1

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]

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

No Iwasawa invariant data is available for this curve.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 5 and 10.
Its isogeny class 369600hd consists of 4 curves linked by isogenies of degrees dividing 10.

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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:

 $[K:\Q]$ $E(K)_{\rm tors}$ Base change curve $K$ $2$ $$\Q(\sqrt{15})$$ $$\Z/2\Z \times \Z/2\Z$$ Not in database $2$ $$\Q(\sqrt{2})$$ $$\Z/10\Z$$ Not in database $4$ 4.4.142296000.4 $$\Z/4\Z$$ Not in database $4$ $$\Q(\sqrt{2}, \sqrt{15})$$ $$\Z/2\Z \times \Z/10\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/2\Z \times \Z/4\Z$$ Not in database $8$ Deg 8 $$\Z/6\Z$$ Not in database $8$ Deg 8 $$\Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/8\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/6\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/2\Z \times \Z/20\Z$$ Not in database $16$ Deg 16 $$\Z/30\Z$$ Not in database $20$ 20.0.1505680748169532571648000000000000000.2 $$\Z/10\Z$$ 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.