Properties

Label 3150bq1
Conductor $3150$
Discriminant $24004512000$
j-invariant \( \frac{461889917}{263424} \)
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([1, -1, 1, -725, -723])
 
gp: E = ellinit([1, -1, 1, -725, -723])
 
magma: E := EllipticCurve([1, -1, 1, -725, -723]);
 

\(y^2+xy+y=x^3-x^2-725x-723\)  Toggle raw display

Mordell-Weil group structure

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

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(-7, 66\right)\)  Toggle raw display
$\hat{h}(P)$ ≈  $0.10214044894327982428775931434$

Torsion generators

sage: E.torsion_subgroup().gens()
 
gp: elltors(E)
 
magma: TorsionSubgroup(E);
 

\( \left(-1, 0\right) \)  Toggle raw display

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(-25, 48\right) \), \( \left(-25, -24\right) \), \( \left(-21, 80\right) \), \( \left(-21, -60\right) \), \( \left(-7, 66\right) \), \( \left(-7, -60\right) \), \( \left(-1, 0\right) \), \( \left(29, 30\right) \), \( \left(29, -60\right) \), \( \left(35, 108\right) \), \( \left(35, -144\right) \), \( \left(63, 416\right) \), \( \left(63, -480\right) \), \( \left(119, 1200\right) \), \( \left(119, -1320\right) \), \( \left(323, 5616\right) \), \( \left(323, -5940\right) \), \( \left(749, 20100\right) \), \( \left(749, -20850\right) \)  Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 3150 \)  =  $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $24004512000 $  =  $2^{8} \cdot 3^{7} \cdot 5^{3} \cdot 7^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{461889917}{263424} \)  =  $2^{-8} \cdot 3^{-1} \cdot 7^{-3} \cdot 773^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $0.68158960476303138409023049983\dots$
Stable Faltings height: $-0.27007601767954855525758195194\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.10214044894327982428775931434\dots$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.99562022493029043236095936398\dots$
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: $ 192 $  = $ 2^{3}\cdot2^{2}\cdot2\cdot3 $
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) $ ≈ $ 4.8812686440666770093302008825507597076 $

Modular invariants

Modular form   3150.2.a.bk

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);
 

\( q + q^{2} + q^{4} + q^{7} + q^{8} - 2q^{11} - 6q^{13} + q^{14} + q^{16} + 4q^{17} - 6q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 3072
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 4 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$(j)_{-}$
$2$ $8$ $I_{8}$ Split multiplicative -1 1 8 8
$3$ $4$ $I_1^{*}$ Additive -1 2 7 1
$5$ $2$ $III$ Additive -1 2 3 0
$7$ $3$ $I_{3}$ Split multiplicative -1 1 3 3

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

$p$-adic regulators

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

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 split add add split ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary ordinary
$\lambda$-invariant(s) 2 - - 2 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

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 $d=$ 2.
Its isogeny class 3150bq consists of 2 curves linked by isogenies of degree 2.

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:

$[K:\Q]$ $K$ $E(K)_{\rm tors}$ Base change curve
$2$ \(\Q(\sqrt{105}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$4$ 4.0.42000.1 \(\Z/4\Z\) Not in database
$8$ 8.0.777924000000.16 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.4.21441530250000.3 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.2.44286750000.4 \(\Z/6\Z\) 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

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.