Properties

Label 930.n1
Conductor $930$
Discriminant $1.357\times 10^{12}$
j-invariant \( \frac{28379906689597370652529}{1357352437500} \)
CM no
Rank $0$
Torsion structure \(\Z/{2}\Z\)

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Show commands: Magma / Pari/GP / SageMath

Minimal Weierstrass equation

sage: E = EllipticCurve([1, 0, 0, -635471, -195033699])
 
gp: E = ellinit([1, 0, 0, -635471, -195033699])
 
magma: E := EllipticCurve([1, 0, 0, -635471, -195033699]);
 

\(y^2+xy=x^3-635471x-195033699\)  Toggle raw display

Mordell-Weil group structure

$\Z/{2}\Z$

Torsion generators

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

\( \left(-\frac{1841}{4}, \frac{1841}{8}\right) \)  Toggle raw display

Integral points

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

None

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 930 \)  =  $2 \cdot 3 \cdot 5 \cdot 31$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $1357352437500 $  =  $2^{2} \cdot 3^{6} \cdot 5^{6} \cdot 31^{3} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( \frac{28379906689597370652529}{1357352437500} \)  =  $2^{-2} \cdot 3^{-6} \cdot 5^{-6} \cdot 31^{-3} \cdot 43^{3} \cdot 223^{3} \cdot 3181^{3}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $1.8053186337294823102425174340\dots$
Stable Faltings height: $1.8053186337294823102425174340\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $0$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $1$
sage: E.period_lattice().omega()
 
gp: E.omega[1]
 
magma: RealPeriod(E);
 
Real period: $0.16909299571714102222658335515\dots$
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: $ 72 $  = $ 2\cdot( 2 \cdot 3 )\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) $ ≈ $ 3.0436739229085384000785003926733207514 $

Modular invariants

Modular form   930.2.a.n

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);
 

\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + q^{8} + q^{9} - q^{10} + q^{12} - 4q^{13} + 2q^{14} - q^{15} + q^{16} + 6q^{17} + q^{18} + 8q^{19} + O(q^{20}) \)  Toggle raw display

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

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 8640
$ \Gamma_0(N) $-optimal: no
Manin constant: 1

Local data

This elliptic curve is 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$(j)_{-}$
$2$ $2$ $I_{2}$ Split multiplicative -1 1 2 2
$3$ $6$ $I_{6}$ Split multiplicative -1 1 6 6
$5$ $2$ $I_{6}$ Non-split multiplicative 1 1 6 6
$31$ $3$ $I_{3}$ 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 $\GL(2,\Z_\ell)$ 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
$3$ 3B.1.2 3.8.0.2

$p$-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 31
Reduction type split split nonsplit split
$\lambda$-invariant(s) 8 1 0 1
$\mu$-invariant(s) 0 1 0 0

All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 2, 3 and 6.
Its isogeny class 930.n consists of 4 curves linked by isogenies of degrees dividing 6.

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
$2$ \(\Q(\sqrt{31}) \) \(\Z/2\Z \times \Z/2\Z\) Not in database
$2$ \(\Q(\sqrt{-3}) \) \(\Z/6\Z\) Not in database
$3$ 3.1.108.1 \(\Z/6\Z\) Not in database
$4$ 4.0.27900.2 \(\Z/4\Z\) Not in database
$4$ \(\Q(\sqrt{-3}, \sqrt{31})\) \(\Z/2\Z \times \Z/6\Z\) Not in database
$6$ 6.0.34992.1 \(\Z/3\Z \times \Z/6\Z\) Not in database
$6$ 6.2.5559715584.1 \(\Z/2\Z \times \Z/6\Z\) Not in database
$8$ 8.4.51120212025600.4 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.11968832160000.12 \(\Z/2\Z \times \Z/4\Z\) Not in database
$8$ 8.0.778410000.3 \(\Z/12\Z\) Not in database
$12$ Deg 12 \(\Z/6\Z \times \Z/6\Z\) Not in database
$12$ Deg 12 \(\Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/8\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$16$ Deg 16 \(\Z/2\Z \times \Z/12\Z\) Not in database
$18$ 18.0.375399958942239013318106246131687500000000.1 \(\Z/18\Z\) 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.